Abstract

The Comment by Kowalczyk et al. [J. Opt. Soc. Am. A 20, 2390 (2003)] on our paper on sampling expansions for three-dimensional light amplitude distributions in the vicinity of an axial image point [J. Opt. Soc. Am. A 14, 2962 (1997)] presents a “new” derivation of the axial sampling expansion (ASE), pointing out some “faults” in our derivation and that of Arsenault and Boivin [J. Appl. Phys. 38, 3988 (1967)], a derivation of which we were not aware. Finally, it also summarizes work done in generalizing the ASE to the case of nonrotationally symmetric pupil functions. The purpose of this reply is to clarify some relevant issues concerning the derivation of this kind of expansion, since, judging from some ideas expressed in their Comment, we believe these issues were left insufficiently clear in our paper. In doing this, we derive some additional sampling expansions.

© 2003 Optical Society of America

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References

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  1. 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). To avoid multiple references to this article, in the present work we shall often refer to it as “our paper.”
    [CrossRef]
  2. A. W. Lohmann, “Optical information processing” (Lecture Notes), 2nd ed. (Erlangen, Germany, 1978), p. 21. Available from the author at Telecommunications Institute, Signal Processing and Multimedia Communications, Universitaet Erlangen-Nuremberg, Cauerstrasse 7/VI, 91058 Erlangen, Germany. Phone, 49-9131-852-7664; e-mail, Lohmann@LNT.de.
  3. A. W. Lohmann, D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [CrossRef] [PubMed]
  4. M. Kowalczyk, T. Cichocki, M. Martı́nez-Corral, L. Muñoz-Escrivá, “Sampling expansions for three-dimensional light amplitude distribution in the vicinity of an axial image point: comment,” J. Opt. Soc. Am. A 20, 2390–2392 (2003).
    [CrossRef]
  5. 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]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 23.
  7. V. Mangulis, Handbook of Series for Scientists and Engineers (Academic, New York, 1966), p. 29, Eqs. (16), (17), and (18).
  8. Staff of the Bateman Manuscript Project, Higher Transcendental Functions (Krieger, Malabar, Fla., 1970), Vol. II, pp. 68–72.
  9. F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), Chap. I.
  10. A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Les Presses de L’Université Laval, Québec, Canada, 1964), p. 198, Eq. II.C-37.
  11. B. Dossier, “Recherches sur l’apodisation des images optiques,” (doctoral thesis) (Éditions de la revue d’Optique, Paris, 1954).
  12. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 7.
  13. M. Martı́nez-Corral, L. Muñoz-Escrivá, M. Kowalczyk, T. Cichocki, “One-dimensional iterative algorithm for three-dimensional point-spread function engineering,” Opt. Lett. 26, 1861–1863 (2001).
    [CrossRef]

2003 (1)

2001 (1)

1997 (1)

1967 (2)

A. W. Lohmann, D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
[CrossRef] [PubMed]

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]

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]

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Les Presses de L’Université Laval, Québec, Canada, 1964), p. 198, Eq. II.C-37.

Bowman, F.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), Chap. I.

Cichocki, T.

Dossier, B.

B. Dossier, “Recherches sur l’apodisation des images optiques,” (doctoral thesis) (Éditions de la revue d’Optique, Paris, 1954).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 23.

Kowalczyk, M.

Landgrave, J. E. A.

Lohmann, A. W.

A. W. Lohmann, D. P. Paris, “Variable Fresnel zone pattern,” Appl. Opt. 6, 1567–1570 (1967).
[CrossRef] [PubMed]

A. W. Lohmann, “Optical information processing” (Lecture Notes), 2nd ed. (Erlangen, Germany, 1978), p. 21. Available from the author at Telecommunications Institute, Signal Processing and Multimedia Communications, Universitaet Erlangen-Nuremberg, Cauerstrasse 7/VI, 91058 Erlangen, Germany. Phone, 49-9131-852-7664; e-mail, Lohmann@LNT.de.

Mangulis, V.

V. Mangulis, Handbook of Series for Scientists and Engineers (Academic, New York, 1966), p. 29, Eqs. (16), (17), and (18).

Marti´nez-Corral, M.

Muñoz-Escrivá, L.

Paris, D. P.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 7.

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. A (2)

Opt. Lett. (1)

Other (8)

A. W. Lohmann, “Optical information processing” (Lecture Notes), 2nd ed. (Erlangen, Germany, 1978), p. 21. Available from the author at Telecommunications Institute, Signal Processing and Multimedia Communications, Universitaet Erlangen-Nuremberg, Cauerstrasse 7/VI, 91058 Erlangen, Germany. Phone, 49-9131-852-7664; e-mail, Lohmann@LNT.de.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 23.

V. Mangulis, Handbook of Series for Scientists and Engineers (Academic, New York, 1966), p. 29, Eqs. (16), (17), and (18).

Staff of the Bateman Manuscript Project, Higher Transcendental Functions (Krieger, Malabar, Fla., 1970), Vol. II, pp. 68–72.

F. Bowman, Introduction to Bessel Functions (Dover, New York, 1958), Chap. I.

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Les Presses de L’Université Laval, Québec, Canada, 1964), p. 198, Eq. II.C-37.

B. Dossier, “Recherches sur l’apodisation des images optiques,” (doctoral thesis) (Éditions de la revue d’Optique, Paris, 1954).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 7.

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Equations (15)

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G(δ, v)=201F(ρ2)exp(-i2πδρ2)J0(vρ)ρdρ,
K(ρ;δ, v)=exp(-i2πδρ2)J0(vρ)
=K(ρ;δ, 0)K(ρ;0, v),
g(ρ2)=m=-cm exp(i2πmρ2),0ρ1,
G(δ, v)=m=-G(m, v)fm(a)(δ, 0)
=m=-G(m, v)L(δ-m, 0),
fm(a)(δ, 0)=exp[-iπ(δ-m)]sinc(δ-m),
K(ρ; δ, v)=m=-fm(a)(δ, 0)K(ρ; m, 0)K(ρ;0, v)
=m=-fm(a)(δ, 0)K(ρ; m, v).
J0(vρ)=n=0fn(r)(0, v)J0(αnρ),0<ρ<1,
G(δ, v)=n=0G(δ, αn)fn(r)(0, v).
fn(r)(0, v)=-(-1)n 2αn2αn2-v2sin vv.
fn(r)(0, v)=1J1(αn)2αnαn2-v2J0(v).
fn(r)(0, v)=-1J0(αn)2vαn2-v2J1(v).
G(δ, v)=m=-n=0G(m, αn)fm(a)(δ, 0)fn(r)(0, v).

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