Abstract

Focusing by an aberration-free cylindrical lens is analyzed in the paraxial Fresnel and Debye approximations, and expressions are given. Plots are given for the intensity in the focal region, the defocused optical transfer function (OTF), the generalized OTF, and the ambiguity function and are compared with the case of an aberration-free spherical lens. Nonparaxial lenses are also discussed.

© 2013 Optical Society of America

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  2. E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002).
    [CrossRef]
  3. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
    [CrossRef]
  4. H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
    [CrossRef]
  5. C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35, 1169–1185 (1988).
    [CrossRef]
  6. G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992).
    [CrossRef]
  7. R. Wolleschensky and B. Zimmermann, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
    [CrossRef]
  8. E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef]
  9. J. S. Marsh, “Light distribution near the focus of a two-dimensional lens,” Am. J. Phys. 52, 152–155 (1984).
    [CrossRef]
  10. D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
    [CrossRef]
  11. D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
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    [CrossRef]
  15. Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987).
    [CrossRef]
  16. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, 1972).
  17. G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
    [CrossRef]
  18. J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  19. J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [CrossRef]
  20. C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
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  21. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  22. C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000).
    [CrossRef]
  23. J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
    [CrossRef]
  24. B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956).
    [CrossRef]
  25. C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
    [CrossRef]
  26. A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [CrossRef]
  27. C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  28. C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003).
    [CrossRef]
  29. R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965).
    [CrossRef]
  30. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955).
    [CrossRef]
  31. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  32. L. Mertz, Transformations in Optics (Wiley, 1965).
  33. B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  34. C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
    [CrossRef]
  35. B. R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966).
    [CrossRef]
  36. C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
    [CrossRef]
  37. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  38. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  39. K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  40. M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
    [CrossRef]
  41. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  42. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
    [CrossRef]
  43. C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
    [CrossRef]
  44. K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
    [CrossRef]
  45. C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001).
    [CrossRef]
  46. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

2007 (1)

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

2006 (3)

R. Wolleschensky and B. Zimmermann, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

2004 (1)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
[CrossRef]

2003 (1)

C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003).
[CrossRef]

2002 (1)

E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002).
[CrossRef]

2001 (1)

C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001).
[CrossRef]

2000 (2)

C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
[CrossRef]

C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000).
[CrossRef]

1999 (1)

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

1998 (1)

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
[CrossRef]

1997 (1)

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

1996 (1)

M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
[CrossRef]

1995 (1)

E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef]

1993 (3)

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170, 229–236 (1993).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef]

1992 (2)

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992).
[CrossRef]

1991 (1)

C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
[CrossRef]

1988 (1)

C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35, 1169–1185 (1988).
[CrossRef]

1987 (2)

Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987).
[CrossRef]

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

1984 (2)

Y. Li and 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]

J. S. Marsh, “Light distribution near the focus of a two-dimensional lens,” Am. J. Phys. 52, 152–155 (1984).
[CrossRef]

1983 (1)

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1981 (3)

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1974 (1)

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

1971 (1)

J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
[CrossRef]

1967 (2)

A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
[CrossRef]

1966 (1)

B. R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966).
[CrossRef]

1965 (1)

R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965).
[CrossRef]

1964 (1)

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

1957 (1)

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

1956 (1)

B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956).
[CrossRef]

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, 1972).

Azzam, F.

E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002).
[CrossRef]

Barakat, R.

R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Becker, K.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Boivin, A.

A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Brakenhoff, G. J.

G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Burns, D. H.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170, 229–236 (1993).
[CrossRef]

Cathey, W. T.

E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef]

Connolly, T. J.

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef]

Deininger, K.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Del Bene, F.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
[CrossRef]

Deussing, J. M.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Dodt, H.-U.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Dow, J.

A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

Dowsky, E. R.

E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef]

Eder, M.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Erkkila, J.

J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

Farnell, G.

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
[CrossRef]

B. R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966).
[CrossRef]

Fuchs, E.

E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002).
[CrossRef]

Goodman, J. W.

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

Gu, M.

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef]

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
[CrossRef]

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955).
[CrossRef]

Hua, J.

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

Huisken, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
[CrossRef]

Jaffe, J. S.

E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002).
[CrossRef]

Jähling, N.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001).
[CrossRef]

C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
[CrossRef]

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

Leischner, U.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Li, G.

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

Li, Y.

Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987).
[CrossRef]

Y. Li and 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]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Liu, L.

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Long, R. A.

E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002).
[CrossRef]

Mao, X.

C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35, 1169–1185 (1988).
[CrossRef]

Marsh, J. S.

J. S. Marsh, “Light distribution near the focus of a two-dimensional lens,” Am. J. Phys. 52, 152–155 (1984).
[CrossRef]

Matthews, H. J.

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

Mauch, C.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

McCutchen, C. W.

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, 1965).

Ojeda-Castanada, J.

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Papoulis, A.

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956).
[CrossRef]

Rogers, M.

J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

Schierloh, A.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003).
[CrossRef]

C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001).
[CrossRef]

C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000).
[CrossRef]

C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
[CrossRef]

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
[CrossRef]

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef]

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
[CrossRef]

C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35, 1169–1185 (1988).
[CrossRef]

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

Sheridan, J. T.

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

Spelman, F. A.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170, 229–236 (1993).
[CrossRef]

Spjelkavik, S.

J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

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

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, 1972).

Stelzer, E. H. K.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
[CrossRef]

Swoger, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
[CrossRef]

Török, P.

C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003).
[CrossRef]

van der Mortel, P. G. J.

M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
[CrossRef]

Visscher, K.

G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992).
[CrossRef]

Voie, A. H.

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170, 229–236 (1993).
[CrossRef]

Wittbrodt, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
[CrossRef]

Wolf, E.

Y. Li and 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]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

Wolleschensky, R.

R. Wolleschensky and B. Zimmermann, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

Zemanek, J.

J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
[CrossRef]

Zieglgänsberger, W.

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Zimmermann, B.

R. Wolleschensky and B. Zimmermann, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

Am. J. Phys. (1)

J. S. Marsh, “Light distribution near the focus of a two-dimensional lens,” Am. J. Phys. 52, 152–155 (1984).
[CrossRef]

Appl. Opt. (1)

E. R. Dowsky and W. T. Cathey, “Extended depth of field through wavefront coding,” Appl. Opt. 34, 1859–1866 (1995).
[CrossRef]

Can. J. Phys. (1)

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
[CrossRef]

J. Biomed. Opt. (1)

R. Wolleschensky and B. Zimmermann, “High-speed confocal fluorescence imaging with a novel line scanning microscope,” J. Biomed. Opt. 11, 064011 (2006).
[CrossRef]

J. Microsc. (3)

G. J. Brakenhoff and K. Visscher, “Confocal imaging with bilateral scanning and array detectors,” J. Microsc. 165, 139–146 (1992).
[CrossRef]

A. H. Voie, D. H. Burns, and F. A. Spelman, “Orthogonal-plane fluorescence optical sectioning: three-dimensional imaging of macroscopic biological specimens,” J. Microsc. 170, 229–236 (1993).
[CrossRef]

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

J. Mod. Opt. (2)

C. J. R. Sheppard and X. Mao, “Confocal microscopes with slit apertures,” J. Mod. Opt. 35, 1169–1185 (1988).
[CrossRef]

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2103 (1998).
[CrossRef]

J. Opt. Soc. Am. (7)

B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
[CrossRef]

B. R. Frieden, “Longitudinal image formation,” J. Opt. Soc. Am. 56, 1495–1501 (1966).
[CrossRef]

J. Erkkila and M. Rogers, “Diffracted fields in the focal region of a convergent wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
[CrossRef]

R. Barakat, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965).
[CrossRef]

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

A. Boivin, J. Dow, and E. Wolf, “Enery flow in the neighbourhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

J. Opt. Soc. Am. A (10)

C. J. R. Sheppard and H. J. Matthews, “Imaging in high aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard and P. Török, “Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003).
[CrossRef]

C. J. R. Sheppard and K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
[CrossRef]

Y. Li and 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]

Y. Li, “Three-dimensional intensity distribution in low Fresnel number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Approximation to the three-dimensional optical transfer function,” J. Opt. Soc. Am. A 8, 692–694 (1991).
[CrossRef]

M. J. Bastiaans and P. G. J. van der Mortel, “Wigner distrubution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
[CrossRef]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[CrossRef]

K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wavefronts,” J. Opt. Soc. Am. A 16, 1838–1844 (1999).
[CrossRef]

Nat. Methods (1)

H.-U. Dodt, U. Leischner, A. Schierloh, N. Jähling, C. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4, 331–336 (2007).
[CrossRef]

Opt. Commun. (6)

D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part I. 2-D system analysis,” Opt. Commun. 263, 171–179 (2006).
[CrossRef]

D. P. Kelly, B. M. Hennelly, J. T. Sheridan, and W. T. Rhodes, “Finite-aperture effects for Fourier transform systems with convergent illumination: part II. 3-D system analysis,” Opt. Commun. 263, 180–188 (2006).
[CrossRef]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

J. J. Stamnes and S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castanada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Opt. Express (1)

E. Fuchs, J. S. Jaffe, R. A. Long, and F. Azzam, “Thin laser sheet microscope for microbial oceanography,” Opt. Express 10, 145–154 (2002).
[CrossRef]

Opt. Lett. (1)

C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000).
[CrossRef]

Optik (1)

C. J. R. Sheppard and K. G. Larkin, “The three-dimensional transfer function and phase space mappings,” Optik 112, 189–192 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409–1412 (1993).
[CrossRef]

Proc. Phys. Soc. B (1)

B. Richards and E. Wolf, “The Airy pattern in systems of high angular aperture,” Proc. Phys. Soc. B 69, 854–856 (1956).
[CrossRef]

Proc. R. Soc. Lond Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond Ser. A 231, 91–103 (1955).
[CrossRef]

Science (1)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305, 1007–1009 (2004).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

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

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, 3rd ed. (Dover, 1972).

L. Mertz, Transformations in Optics (Wiley, 1965).

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

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Figures (12)

Fig. 1.
Fig. 1.

Contours of constant intensity in the focal region of (a) an aberration-free cylindrical lens and (b) a spherical lens, in the paraxial Debye regime.

Fig. 2.
Fig. 2.

Intensity in the focal region of (a) an aberration-free cylindrical lens and (b) a spherical lens, in the paraxial Debye regime shown as 3D plots.

Fig. 3.
Fig. 3.

Intensity (a) in the focal plane and (b) along the shadow edge, v=u, of a cylindrical lens (solid curve) and a spherical lens (dotted curve).

Fig. 4.
Fig. 4.

(a) Intensity along the optical axis of a cylindrical lens (solid curve) and a spherical lens (dotted curve). (b) The axial phase variation, with kz suppressed, for a cylindrical lens (solid curve) and a spherical lens (dotted line).

Fig. 5.
Fig. 5.

Fraction of energy E with a region |v|vd for a cylindrical lens (solid curve) and a spherical lens (dotted curve).

Fig. 6.
Fig. 6.

Time-averaged electric energy density in the focal region of a cylindrical lens of angular semiaperture 60° for (a) s polarization and (b) p polarization in the Debye regime shown as 3D plots.

Fig. 7.
Fig. 7.

Time-averaged electric energy density (a) in the focal plane and (b) along the axis, of a cylindrical lens of angular semiaperture 60° in the Debye regime. s polarization is shown as solid curves and p polarization as dotted curves.

Fig. 8.
Fig. 8.

(a) Flow curves of the Poynting vector for either an s- or a p-polarized cylindrical lens of angular semiaperture 60°. (b) Closeup of a region containing two singularities and a vortex.

Fig. 9.
Fig. 9.

Defocused OTF for a cylindrical lens (solid curves) and a spherical lens (dotted curves). The curves are shown for u=0, 4, 8, 12, 16.

Fig. 10.
Fig. 10.

Generalized (2D) OTF G(m,s) for a cylindrical lens. The axes are m (normalized transverse spatial frequency) and s (normalized axial spatial frequency). Reproduced from [45].

Fig. 11.
Fig. 11.

Axial OTF for a cylindrical lens.

Fig. 12.
Fig. 12.

Ambiguity function for a cylindrical lens. The horizontal axis is m and the vertical axis is v=um/2. Gray levels range from 0.25 (black) to 1 (white).

Equations (28)

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

U(x,z)=1λzeiπ/4eikzexp(ikx22z)P(x)exp[ikx22(1f1z)]exp(ikxxz)dx,
v=kfxzsinα=kxaz,
u=ka2(1f1z)=(fz)kδzsin2α,
U(v,u)=Neiπ/4eikzexp(iv24πN)11P(ξ)exp(iuξ22)exp(ivξ)dξ.
U(v,u)=Neiπ/4eikzexp(iv24πN)exp(iv22u)πu[F*(u+v2u)+F*(uv2u)],
F(z)=C(z)+iS(z)
F(z)=(1+i)2erf[π2(1i)z].
U(0,0)=2N0eiπ/4eikf.
U(v,u)=12NN0eik(zf)exp[iv22u(N0N)]πu[F*(u+vπu)+F*(uvπu)].
U(0,u)=NN0eik(zf)πuF*(uπ).
U(v,0)=exp(iv24πN0)(sinvv).
E(vd)=2π[Si(2vd)sin2vdvd],
Ey=ααcos1/2θexp[ik(xsinθ+zcosθ)]dθ,
Ex=ααcos3/2θexp[ik(xsinθ+zcosθ)]dθ,Ez=ααcos1/2θsinθexp[ik(xsinθ+zcosθ)]dθ.
u=4kδzsin2(α2),
z=u4kfsin2(α/2){1(u/4kf)1[u/4kfsin2(α/2)]},
P(ξ)=P0(ξ)exp(12iuξ2),ξ<1,
c(m)=c0(m)exp(12ium2),m<1,
C(m,u)=c(m+m/2)c*(mm/2)dm|c(m)|2dm.
C(m,u)=sin[(1m/2)mu]mu,|m|2.
C(m,u)=sin(mu)mu,|m|2.
Π(m,s)=c0(m)δ(sm22),|m|<1,
caxial(s)=11δ(sm22)dm=12|s|,0<s<12.
G(m,s)=C(m,u)exp(ius)du.
G(m,s)=12|m|,|s|<|m|(2|m|),|m|<2.
L(s)=12ln(1+12|s|112|s|)=ln(1+12|s|2|s|),|s|12.
A(m,v)=sin[(2|m|)v]2v,|m|2.
W(m,v)=sin[2(1|m|)v]2πv,|m1|.

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