Abstract

In a recent paper by P. Wang et al. [J. Opt. Soc. Am. A 15, 684 (1998)] an analytic expression for calculating Fresnel diffraction is presented. Several mathematical problems with this analysis have been observed. In this comment, corrections to the analysis of Wang et al. are offered as well as an alternative view of this analysis. New formulas based on series of exponential polynomials that are everywhere convergent are derived. These new formulas eliminate the need to split the computational problem into two distinct regions. New series representations for two of the most important combinations of the Lommel functions of two variables are offered.

© 1999 Optical Society of America

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References

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  1. P. Wang, Y. Xu, W. Wang, Z. Wang, “Analytic expression for Fresnel diffraction,” J. Opt. Soc. Am. A 15, 684–688 (1998).
    [CrossRef]
  2. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1958), Chap. 16.
  3. J. Walker, The Analytical Theory of Light (Cambridge U. Press, London, 1904).
  4. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [CrossRef]
  5. J. Spanier, K. Oldham, An Atlas of Functions (Hemisphere, New York, 1987).
  6. P. L. Overfelt, “Partial summation of the Wang series,” (Naval Air Warfare Center Weapons Division, China Lake, Calif., July1998).

1998 (1)

1991 (1)

Kenney, C. S.

Oldham, K.

J. Spanier, K. Oldham, An Atlas of Functions (Hemisphere, New York, 1987).

Overfelt, P. L.

Spanier, J.

J. Spanier, K. Oldham, An Atlas of Functions (Hemisphere, New York, 1987).

Walker, J.

J. Walker, The Analytical Theory of Light (Cambridge U. Press, London, 1904).

Wang, P.

Wang, W.

Wang, Z.

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1958), Chap. 16.

Xu, Y.

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

Other (4)

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1958), Chap. 16.

J. Walker, The Analytical Theory of Light (Cambridge U. Press, London, 1904).

J. Spanier, K. Oldham, An Atlas of Functions (Hemisphere, New York, 1987).

P. L. Overfelt, “Partial summation of the Wang series,” (Naval Air Warfare Center Weapons Division, China Lake, Calif., July1998).

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Equations (20)

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S1=n=1-i ρManJn2Nπ ρa
S2=n=1-i MaρnJn2Nπ ρa
RHS=expρ2M2a2-1e-1
S1=-iV1-2πMN, 2πNρa-V2-2πMN, 2πNρa,ρMa<1exp-iπMN1+ρ2M2a2-iU1-2πMN, 2πNρa-U0-2πMN, 2πNρa,ρMa>1.
S2=exp-iπMN1+ρ2M2a2+iV1-2πMN, 2πNρa-V0-2πMN, 2πNρa,ρMa<1iU1-2πMN, 2πNρa-U2-2πMN, 2πNρa,ρMa>1.
p=-iV1(w, z)-V2(w, z)expiw2+z22w+iV1(w, z)-V0(w, z),zw<1
p=expiw2+z22w-iU1(w, z)-U0(w, z)iU1(w, z)-U2(w, z), zw>1.
en(x)=1+x+x22!+x33! +xnn!
e(x)=exp(x),
p=k=0 (-iπMN)kk!exp-iπMNρMa2-ek-iπMNρMa2k=0 (-iπMN)ρMa2kk![exp(-iπMN)-ek(-iπMN)].
pexp-iπMNρMa2-1exp(-iπMN)-1,
D=11+p
p=S1S2,
D=S2S1+S2.
Dexp(-iπMN)-1exp(-iπMN)+exp-iπMNρMa2-2,
S1+S2=u-J02Nπρa
S1+S2=exp-iπMN1+ρ2M2a2-J02Nπρa.
D=k=0 -iπMNρMa2kk![exp(-iπMN)-ek(-iπMN)]exp-iπMN1+ρ2M2a2-J0(2Nπρ/a).
iU1(w, z)-U2(w, z)=k=0 [(iz2)/(2w)]kk!expiw2-ekiw2,
-iV1(w, z)-V2(w, z)=k=0 [(iw)/2]kk!expiz22w-ekiz22w.

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