## Generalized Jinc functions and their application to focusing and diffraction of circular apertures

JOSA A, Vol. 20, Issue 4, pp. 661-667 (2003)

http://dx.doi.org/10.1364/JOSAA.20.000661

Acrobat PDF (162 KB)

### Abstract

A family of generalized Jinc functions is defined and analyzed. The zero-order one is just the traditional Jinc function. In terms of these functions, series-form expressions are presented for the Fresnel diffraction of a circular aperture illuminated by converging spherical waves or plane waves. The leading term is nothing but the Airy formula for the Fraunhofer diffraction of circular apertures, and those high-order terms are directly related to those high-order Jinc functions. The truncation error of the retained terms is also analytically investigated. We show that, for the illumination of a converging spherical wave, the first 19 terms are sufficient for describing the three-dimensional field distribution in the whole focal region.

© 2003 Optical Society of America

**OCIS Codes**

(000.3870) General : Mathematics

(050.1220) Diffraction and gratings : Apertures

(050.1940) Diffraction and gratings : Diffraction

(050.1960) Diffraction and gratings : Diffraction theory

(220.2560) Optical design and fabrication : Propagating methods

**Citation**

Qing Cao, "Generalized Jinc functions and their application to focusing and diffraction of circular apertures," J. Opt. Soc. Am. A **20**, 661-667 (2003)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-4-661

Sort: Year | Journal | Reset

### References

- M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), Chap. 8.
- J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 2.
- A. E. Siegman, Lasers (Oxford U. Press, Oxford, UK, 1986), Sec. 18.4.
- Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in system of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
- Y. Li, “Three-dimensional intensity distribution in low-Fresnel-number focusing systems,” J. Opt. Soc. Am. A 4, 1349–1353 (1987).
- P. Wang, Y. Xu, W. Wang, and Z. Wang, “Analytical expression for Fresnel diffraction,” J. Opt. Soc. Am. A 15, 684–688 (1998).
- J. C. Heurtley, “Analytical expression for Fresnel diffraction: comment,” J. Opt. Soc. Am. A 15, 2929–2930 (1998).
- P. L. Overfelt and D. J. White, “Analytical expression for Fresnel diffraction: comment,” J. Opt. Soc. Am. A 16, 613–615 (1999).
- Y.-T. Wang, Y. C. Pati, and T. Kailath, “Depth of focus and the moment expansion,” Opt. Lett. 20, 1841–1843 (1995).
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972).
- W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
- J. H. Erkkila and M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
- Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
- J. J. Stamnes and B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
- M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
- Q. Cao and J. Jahns, “Focusing analysis of the pinhole photon sieve: individual far-field model,” J. Opt. Soc. Am. A 19, 2387–2393 (2002).
- L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft x-rays with photon sieve,” Nature (London) 414, 184–188 (2001).
- E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968). The Parseval theorem is the special case f(x, y)= g(x, y) of Theorem IV of this reference.
- W. H. Southwell, “Asymptotic solution of the Huygens–Fresnel integral in circular coordinates,” Opt. Lett. 3, 100–102 (1978).
- A. J. E. M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
- J. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.