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]

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).

[Crossref]

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]

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]

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

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]

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

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]

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]

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]

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]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).

[Crossref]

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

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]

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, 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]

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]

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

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]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

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]

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, 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]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).

[Crossref]

D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).

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]

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]

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.

The calculations reported in Refs. 13 and 14 are based on Eq. (B5).