Abstract

Landgrave and Berriel-Valdos presented axial and radial sampling expansions for three-dimensional light amplitude distribution around the Gaussian focal point. [J. Opt. Soc. Am. A 14, 2962 (1997)]. The expansions were obtained under the assumption that the pupil function was rotationally symmetric. We present a new derivation of the axial expansion that does not make use of arbitrary formal assumptions used by Landgrave and Berriel-Valdos and eliminates some faults of the derivation given by Arsenault and Boivin, who published this expansion in 1967 [J. Appl. Phys. 38, 3988 (1967)]. We also discuss generalizations of the axial expansion to the case of pupils that exhibit no symmetry with respect to the axis considered.

© 2003 Optical Society of America

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References

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  1. M. Martı́nez-Corral, L. Muñoz-Escrivá, M. Kowalczyk, T. Cichocki, “One-dimensional iterative algorithm for 3D point-spread function engineering,” Opt. Lett. 26, 1861–1863 (2001).
    [CrossRef]
  2. J. E. A. Landgrave, L. R. Berriel-Valdos, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point,” J. Opt. Soc. Am. A 14, 2962–2976 (1997).
    [CrossRef]
  3. H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
    [CrossRef]
  4. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  5. B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and application,” Opt. Commun. 28, 35–38 (1979).
    [CrossRef]
  6. P. Andrés, M. Martı́nez-Corral, J. Ojeda-Castañeda, “Off-axis focal shift for rotationally nonsymmetric screens,” Opt. Lett. 18, 1290–1292 (1993).
    [CrossRef] [PubMed]
  7. R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
    [CrossRef]
  8. S. C. Som, “Multiple reproduction of space-limited functions by an optical technique,” Phys. Lett. A 29, 600–601 (1969).
    [CrossRef]
  9. L. D. Dickson, “Characteristics of a propagating Gaussian beam,” Appl. Opt. 9, 1854–1861 (1970).
    [CrossRef] [PubMed]
  10. S. C. Som, “Simultaneous multiple reproduction of space-limited functions by sampling of spatial frequencies,” J. Opt. Soc. Am. 60, 1628–1634 (1970).
    [CrossRef]
  11. M. Novotny, “Foci of axially symmetrical filters,” Opt. Acta 20, 217–232 (1973).
    [CrossRef]
  12. M. Novotny, “New series representation of Fresnel diffraction field of axially symmetric filters,” Opt. Acta 24, 551–565 (1977).
    [CrossRef]
  13. J. Ojeda-Castañeda, “Focus-error operator and related special functions,” J. Opt. Soc. Am. 73, 1042–1047 (1983).
    [CrossRef]
  14. P. R. King, “The design of diffractive surface relief lenses with more than one focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 312–314 (1985).
  15. M. Dragoman, “Synthesis of circular apertures via the radial sampling theorem,” Rev. Roum. Sci. Techn. Electrotechn. Energ. 32, 329–335 (1987).

2001 (1)

1997 (1)

1996 (1)

1993 (1)

1987 (1)

M. Dragoman, “Synthesis of circular apertures via the radial sampling theorem,” Rev. Roum. Sci. Techn. Electrotechn. Energ. 32, 329–335 (1987).

1985 (1)

P. R. King, “The design of diffractive surface relief lenses with more than one focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 312–314 (1985).

1983 (1)

1979 (1)

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and application,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

1977 (1)

M. Novotny, “New series representation of Fresnel diffraction field of axially symmetric filters,” Opt. Acta 24, 551–565 (1977).
[CrossRef]

1973 (1)

M. Novotny, “Foci of axially symmetrical filters,” Opt. Acta 20, 217–232 (1973).
[CrossRef]

1970 (2)

1969 (1)

S. C. Som, “Multiple reproduction of space-limited functions by an optical technique,” Phys. Lett. A 29, 600–601 (1969).
[CrossRef]

1967 (1)

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

Andrés, P.

Arsenault, H.

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

Berriel-Valdos, L. R.

Boivin, A.

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Cichocki, T.

Colombeau, B.

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and application,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

Dickson, L. D.

Dragoman, M.

M. Dragoman, “Synthesis of circular apertures via the radial sampling theorem,” Rev. Roum. Sci. Techn. Electrotechn. Energ. 32, 329–335 (1987).

Froehly, C.

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and application,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

King, P. R.

P. R. King, “The design of diffractive surface relief lenses with more than one focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 312–314 (1985).

Kowalczyk, M.

Landgrave, J. E. A.

Marti´nez-Corral, M.

Muñoz-Escrivá, L.

Novotny, M.

M. Novotny, “New series representation of Fresnel diffraction field of axially symmetric filters,” Opt. Acta 24, 551–565 (1977).
[CrossRef]

M. Novotny, “Foci of axially symmetrical filters,” Opt. Acta 20, 217–232 (1973).
[CrossRef]

Ojeda-Castañeda, J.

Piestun, R.

Shamir, J.

Som, S. C.

S. C. Som, “Simultaneous multiple reproduction of space-limited functions by sampling of spatial frequencies,” J. Opt. Soc. Am. 60, 1628–1634 (1970).
[CrossRef]

S. C. Som, “Multiple reproduction of space-limited functions by an optical technique,” Phys. Lett. A 29, 600–601 (1969).
[CrossRef]

Spektor, B.

Vampouille, M.

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and application,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

Acta Polytech. Scand. Appl. Phys. Ser. (1)

P. R. King, “The design of diffractive surface relief lenses with more than one focus,” Acta Polytech. Scand. Appl. Phys. Ser. 149, 312–314 (1985).

Appl. Opt. (1)

J. Appl. Phys. (1)

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (2)

M. Novotny, “Foci of axially symmetrical filters,” Opt. Acta 20, 217–232 (1973).
[CrossRef]

M. Novotny, “New series representation of Fresnel diffraction field of axially symmetric filters,” Opt. Acta 24, 551–565 (1977).
[CrossRef]

Opt. Commun. (1)

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and application,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

S. C. Som, “Multiple reproduction of space-limited functions by an optical technique,” Phys. Lett. A 29, 600–601 (1969).
[CrossRef]

Rev. Roum. Sci. Techn. Electrotechn. Energ. (1)

M. Dragoman, “Synthesis of circular apertures via the radial sampling theorem,” Rev. Roum. Sci. Techn. Electrotechn. Energ. 32, 329–335 (1987).

Other (1)

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

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Figures (1)

Fig. 1
Fig. 1

Geometry of the problem. A pupil filter, or diffracting aperture, F(ρ), is illuminated by a spherical wave that converges to the focal point F. In Ref. 6 it is shown that the amplitude distribution along the AB segment is proportional to the one-dimensional FT of the zero-order circular-harmonic component of F(ρ, θ), calculated for the point A, which is the center of the circular-harmonic expansion. Point B is located in the observation plane, which is at a distance z from the image or focal plane.

Equations (10)

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G(δ, v)=m=-G(m, 0)L(δ-m, v),
G(y, z)=n=-G(-4nπ, 0)O(y+4nπ, z),
pT(r)=1,-T/2<r<T/20,otherwise,
G(δ, v)=201F(ρ2)exp(-i2πδρ2)J0(ρv)ρdρ.
exp(-i2πδρ2)J0(vρ)=m=-fm(a)(δ, v)exp(-i2πmρ2),
p(ρ)=1,0<ρ<10,otherwise,
G(δ, v)=0p(ρ)F(ρ)exp(-i2πδρ)J0(vρ1/2)dρ.
Fp(ρ)=p(ρ)m=-+Cm exp(i2πmρ),
Cm=01F(ρ)exp(-2πimρ)dρ=G(m, 0).
G(δ,v)=0p(ρ)m=-G(m, 0)exp(2πimρ)×exp(-i2πδρ)J0(vρ1/2)dρ=m=-G(m, 0)01 exp[-2πi(δ-m)ρ]×J0(vρ1/2)dρ=2m=-G(m, 0)01 exp[-2πi(δ-m)ρ2]×J0(vρ)ρdρ=m=-G(m, 0)L(δ-m, v),

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