## Waveletlike basis function approach to the propagation of paraxial beams

JOSA A, Vol. 17, Issue 6, pp. 1043-1047 (2000)

http://dx.doi.org/10.1364/JOSAA.17.001043

Enhanced HTML Acrobat PDF (138 KB)

### Abstract

A semianalytical method is described for calculating the diffraction integral for paraxial propagation through an optical system. The field at the input plane is represented by a linear superposition of nearly Gaussian basis functions that keep a simple analytical form under *ABCD* propagation. The coefficients of the superposition are obtained by a numerical fit. The flexibility of the basis functions makes the method well suited to dealing with sharp local variations of the input field.

© 2000 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(050.1970) Diffraction and gratings : Diffractive optics

(260.1960) Physical optics : Diffraction theory

**History**

Original Manuscript: September 14, 1999

Revised Manuscript: March 7, 2000

Manuscript Accepted: March 7, 2000

Published: June 1, 2000

**Citation**

Robert M. Potvliege, "Waveletlike basis function approach to the propagation of paraxial beams," J. Opt. Soc. Am. A **17**, 1043-1047 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-6-1043

Sort: Year | Journal | Reset

### References

- P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969). [CrossRef]
- P. Baues, “The connection of geometrical optics with the propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969). [CrossRef]
- S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]
- A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
- The definition of the transfer matrix adopted here follows that given in Ref. 4.
- I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).
- Since the coefficients of the expansion are not obtained by projection of the input field on the basis functions, the only practical constraint on their normalization is that their maximum value should not be so large, or so small, as to hinder the numerical computations.
- A mathematically rigorous discussion of the completeness of pseudowavelet bases is outside the scope of this work. It is clear, however, that any smoothly varying field distribution can be approached at any desirable level of accuracy by a linear superposition of such functions, as is the case with spline bases. A basis containing only pseudowavelets with the same parameter a and with n varying from 0 to N by a step of one is equivalent to a basis of N Laguerre–Gaussian modes. Bases containing pseudowavelets with different values of a, on the other hand, are subsets of overcomplete bases formed by the union of several Laguerre–Gaussian bases.
- Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)]. [CrossRef] [PubMed]
- F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994). [CrossRef]
- V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996). [CrossRef]
- D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).
- B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999). [CrossRef]
- B. Lü, B. Zhang, S. R. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999). [CrossRef]
- The calculations reported in Refs. 13 and 14 are based on Eq. (B5).

## 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.