Abstract

A new superposition scheme for representing flattened Gaussian (FG) beams is proposed. Such a representation, unlike the original proposed by Gori [Opt. Commun. 107, 335 (1994)], is based on an expansion in terms of the so-called elegant Laguerre–Gaussian beams. This new representation allows us to obtain the closed-form expression of a FG beam of any order propagating through a paraxial ABCD optical system by means of a simple recurrence rule, which turns out to be particularly stable even when it is applied to FG beams of very high orders (>104).

© 2001 Optical Society of America

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  1. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  2. C. Palma, V. Bagini, “Expansions of general beams in Gaussian beams,” Opt. Commun. 116, 1–7 (1995).
    [CrossRef]
  3. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation features of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  4. S.-A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
    [CrossRef]
  5. B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
    [CrossRef]
  6. M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
    [CrossRef]
  7. X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
    [CrossRef]
  8. D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated intensity of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).
  9. R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
    [CrossRef]
  10. 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]
  11. B. Lü, S. Luo, “General propagation equation of flattened Gaussian beams,” J. Opt. Soc. Am. A 17, 2001–2004 (2000).
    [CrossRef]
  12. R. Borghi, M. Santarsiero, “Modal decomposition of flat-topped beams produced by multimode stable-cavity lasers,” Opt. Lett. 23, 313–315 (1998).
    [CrossRef]
  13. R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
    [CrossRef]
  14. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  15. C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
    [CrossRef]
  16. R. M. Potvliege, “Waveletlike basis function approach to the propagation of paraxial beams,” J. Opt. Soc. Am. A 17, 1043–1047 (2000).
    [CrossRef]
  17. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
    [CrossRef]
  18. A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
    [CrossRef]
  19. B. Lü, B. Zhang, X. Wang, “Three-dimensional intensity distribution of focused super-Gaussian beams,” Opt. Commun. 126, 1–6 (1996).
    [CrossRef]
  20. S.-A. Amarande, “Approximation of super-Gaussian beams by generalized flattened Gaussian beams,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, D. R. Hall, H. J. Baker, eds., Proc. SPIE3092, 345–348 (1997).
    [CrossRef]
  21. M. Santarsiero, R. Borghi, “On the correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
    [CrossRef]
  22. B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–288 (1999).
  23. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  24. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  25. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  26. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  27. For brevity we limit ourselves to the three-dimensional axisymmetric case, but the extension to the rectangular case is immediate.
  28. As usual, circ(r)is defined as 1 if r≤1and 0 elsewhere, rbeing the radial coordinate in R2.
  29. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6, p. 785.
  30. Note that Eq. (3) differs from the formula quoted in Ref. 29by a factor of n! as the result of the different normalization for the Laguerre polynomial used here, i.e., Ln(0)=1.
  31. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.
  32. H. Ma, B. Lü, “The propagation of complex-argument Laguerre-Gaussian beams,” Optik 111, 273–279 (2000).
  33. A. Siegman, Lasers (University Science, Mill Valley, 1986).
  34. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  35. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  36. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.
  37. M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).
  38. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]

2000 (3)

1999 (5)

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

M. Santarsiero, R. Borghi, “On the correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–288 (1999).

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (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]

1998 (3)

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

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal decomposition of flat-topped beams produced by multimode stable-cavity lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

1997 (2)

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

1996 (4)

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

S.-A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
[CrossRef]

B. Lü, B. Zhang, X. Wang, “Three-dimensional intensity distribution of focused super-Gaussian beams,” Opt. Commun. 126, 1–6 (1996).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation features of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

1995 (1)

C. Palma, V. Bagini, “Expansions of general beams in Gaussian beams,” Opt. Commun. 116, 1–7 (1995).
[CrossRef]

1994 (1)

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

1992 (1)

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

1988 (1)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

1986 (1)

1985 (1)

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1973 (1)

Aiello, D.

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

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

Amarande, S.-A.

S.-A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
[CrossRef]

S.-A. Amarande, “Approximation of super-Gaussian beams by generalized flattened Gaussian beams,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, D. R. Hall, H. J. Baker, eds., Proc. SPIE3092, 345–348 (1997).
[CrossRef]

Ambrosini, D.

Bagini, V.

Borghi, R.

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, R. Borghi, “On the correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

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

R. Borghi, M. Santarsiero, “Modal decomposition of flat-topped beams produced by multimode stable-cavity lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation features of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Deng, X.

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

Fan, D.

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6, p. 785.

Fukumitsu, O.

Gori, F.

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Lavigne, P.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, Y.

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lü, B.

B. Lü, S. Luo, “General propagation equation of flattened Gaussian beams,” J. Opt. Soc. Am. A 17, 2001–2004 (2000).
[CrossRef]

H. Ma, B. Lü, “The propagation of complex-argument Laguerre-Gaussian beams,” Optik 111, 273–279 (2000).

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. 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, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–288 (1999).

B. Lü, B. Zhang, X. Wang, “Three-dimensional intensity distribution of focused super-Gaussian beams,” Opt. Commun. 126, 1–6 (1996).
[CrossRef]

Luo, S.

B. Lü, S. Luo, “General propagation equation of flattened Gaussian beams,” J. Opt. Soc. Am. A 17, 2001–2004 (2000).
[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, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–288 (1999).

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

Ma, H.

H. Ma, B. Lü, “The propagation of complex-argument Laguerre-Gaussian beams,” Optik 111, 273–279 (2000).

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Morin, M.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6, p. 785.

Pacileo, A. M.

Palma, C.

C. Palma, V. Bagini, “Expansions of general beams in Gaussian beams,” Opt. Commun. 116, 1–7 (1995).
[CrossRef]

Parent, A.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Potvliege, R. M.

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

Qiu, Y.

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Saghafi, S.

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

Santarsiero, M.

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

M. Santarsiero, R. Borghi, “On the correspondence between super-Gaussian and flattened Gaussian beams,” J. Opt. Soc. Am. A 16, 188–190 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

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

R. Borghi, M. Santarsiero, “Modal decomposition of flat-topped beams produced by multimode stable-cavity lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation features of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Schirripa Spagnolo, G.

Sheppard, C. J. R.

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

Siegman, A.

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

Siegman, A. E.

Svelto, O.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Takenaka, T.

Vicalvi, S.

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

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

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

Wang, X.

B. Lü, B. Zhang, X. Wang, “Three-dimensional intensity distribution of focused super-Gaussian beams,” Opt. Commun. 126, 1–6 (1996).
[CrossRef]

Wolf, E.

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Yokota, M.

Zauderer, E.

Zhang, B.

B. Lü, B. Zhang, S. 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, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–288 (1999).

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, X. Wang, “Three-dimensional intensity distribution of focused super-Gaussian beams,” Opt. Commun. 126, 1–6 (1996).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

J. Mod. Opt. (1)

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (9)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

X. Deng, Y. Li, D. Fan, Y. Qiu, “Propagation of paraxial circular symmetric beams in a general optical system,” Opt. Commun. 140, 226–230 (1997).
[CrossRef]

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

C. Palma, V. Bagini, “Expansions of general beams in Gaussian beams,” Opt. Commun. 116, 1–7 (1995).
[CrossRef]

S.-A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
[CrossRef]

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[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, B. Zhang, X. Wang, “Three-dimensional intensity distribution of focused super-Gaussian beams,” Opt. Commun. 126, 1–6 (1996).
[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Optik (3)

H. Ma, B. Lü, “The propagation of complex-argument Laguerre-Gaussian beams,” Optik 111, 273–279 (2000).

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

B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–288 (1999).

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Other (12)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

S.-A. Amarande, “Approximation of super-Gaussian beams by generalized flattened Gaussian beams,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, D. R. Hall, H. J. Baker, eds., Proc. SPIE3092, 345–348 (1997).
[CrossRef]

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

For brevity we limit ourselves to the three-dimensional axisymmetric case, but the extension to the rectangular case is immediate.

As usual, circ(r)is defined as 1 if r≤1and 0 elsewhere, rbeing the radial coordinate in R2.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6, p. 785.

Note that Eq. (3) differs from the formula quoted in Ref. 29by a factor of n! as the result of the different normalization for the Laguerre polynomial used here, i.e., Ln(0)=1.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

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

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

M. Born, E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge U. Press, Cambridge, UK, 1999).

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Figures (1)

Fig. 1
Fig. 1

Behavior of the optical intensity transverse profile as a function of the normalized variable r/w0 for a FG beam having a value of w0=1 mm across the plane corresponding to the Fresnel number NF=10. Values of N are (a) 0, (b) 10, (c) 102, (d) 103, (e) 104, and (f) 105. The wavelength is λ=0.5μm.

Equations (51)

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UN(0)(r; w0)=A0fNr(N+1)1/2w0,
fN(ξ)=exp(-ξ2)m=0N1m! ξ2m.
ξ2mm!=n=0m(-1)nmnLn(ξ2),
fN(ξ)=exp(-ξ2)m=0Nn=0m(-1)nmnLn(ξ2)=n=0N(-1)nm=nNmnLn(ξ2)exp(-ξ2)=n=0N(-1)nN+1n+1Ln(ξ2)exp(-ξ2),
UN(0)(r; w0)=A0n=0N(-1)nN+1n+1ψn(0)(r).
ψn(0)(r)=Ln-ik2qN(0) r2expik2qN(0) r2,
ψn(1)(r)=exp(ikl) 1A+B/qN(0)11+B/AqN(0)n×expik2qN(1) r2Ln-ikr22A[AqN(0)+B],
qN(1)=AqN(0)+BCqN(0)+D.
UN(1)(r; w0)=A0exp(ikl) 1A+B/qN(0)×expik2qN(1) r2×n=0NN+1n+1×-11+B/AqN(0)n×Ln-ikr22A[AqN(0)+B].
UN(1)(r; w0)=A0exp(ikl) 1A+B/qN(0)expik2qN(1) r2×GN-11+B/AqN(0), -ikr22A[AqN(0)+B],
GN(t, s)=n=0NN+1n+1tnLn(s).
GN(t, 0)=n=0NN+1n+1tnLn(0)=n=0NN+1n+1tn=m=1N+1N+1mtm-1=1tm=0N+1N+1mtm-1=(1+t)N+1-1t,
 UN(1)(0; w0)=-A0Aexp(ikl)×1-11+B/AqN(0)N+1-1.
ABCD=1z01 10-1f1=1-zfz-1f1,
u=(F/f)(z-f),
UN(1)(0; w0)
=A0F expikf+ufw02
×1-11-[i2(N+1)/u](1+u/F)N+1-1u,
GN(t, s)s=n=0NN+1n+1tndLn(s)ds=n=0NN+1n+1tnLn(s)-n=0NN+1n+1tnLn(1)(s),
n=0NN+1n+1tnLn(1)(s)=(1+t)NLN(1)tst+1,
GN(t, s)s=GN(t, s)-(1+t)NLN(1)tst+1.
GN(t, s)=exp(s)(1+t)N+1-1t-(1+t)N0sexp(-ξ)LN(1)tξt+1dξ,
G0(t, s)=1,
G1(t, s)=-st+(t+2),
GN+1(t, s)=-(1+t)N+1LNtst+1-LN+1tst+1+(2+t)GN(t, s)-(1+t)GN-1(t, s).
t=-11+B/AqN(0),s=-ikr22A[AqN(0)+B],
NF=w02/λz
GN+1(-1, s)=GN(-1, s)-lim0 N+1×LN-s-LN+1-s=GN(-1, s)+lim0N+1LN+1-s=GN(-1, s)+sN+1(N+1)!,
GN(-1, s)=n=0Nsnn!,
GN(t, s)=exp(s)(1+t)N+1-1t-(1+t)N×0sexp(-ξ)LN(1)tξt+1dξ.
IN(β, s)=0sexp(-ξ)LN(1)(βξ)dξ,
I0(β, s)=1-exp(-s),
I1(β, s)=(2-β)[1-exp(-s)]+βs exp(-s),
IN+1(β, s)=0sexp(-ξ)LN+1(1)(βξ)dξ,
IN+1(β, s)=2IN(β, s)-IN-1(β, s)
-βN+10s ξ exp(-ξ)LN(1)(βξ)dξ.
0s ξ exp(-ξ)LN(1)(βξ)dξ
=[-ξ exp(-ξ)LN(1)(βξ)]ξ=0ξ=s+0sexp(-ξ)LN(1)×(βξ)dξ+0s ξ exp(-ξ) ddξ LN(1)(βξ)dξ=-s exp(-s)LN(1)(βs)+IN(β, s)+0s βξ exp(-ξ)LN(1)(βξ)dξ,
0sβξ exp(-ξ)LN(1)(βξ)dξ
=NIN(β, s)-(N+1)IN-1(β, s),
IN+1(β, s)=2IN(β, s)-IN-1(β, s)+βs exp(-s)N+1 LN(1)×(βs)+β[IN(β, s)-IN-1(β, s)]=exp(-s)[LN(βs)-LN+1(βs)]+(2-β)IN(β, s)-(1-β)IN-1(β, s),
GN(t, s)
=exp(s)(1+t)N+1-1t-(1+t)NINtt+1, s,
GN+1(t, s)=exp(s)(1+t)N+2-1t-(1+t)N+1×exp(-s)LNtst+1-LN+1tst+1-(1+t)N(2t)INtt+1, s-IN-1tt+1, s=-(1+t)N+1LNtst+1-LN+1tst+1+exp(s)(1+t)N+2-1t-(1+t)NIN×tt+1, s+exp(s)(1+t)N+1-(1+t)N(1+t)INtt+1, s-IN-1tt+1, s.
(1+t)N+1INtt+1, s
=1+tt [(1+t)N+1-1]-(1+t)
×exp(-s)GN(t, s),
(1+t)NIN-1tt+1, s
=1+tt [(1+t)N-1]-(1+t)
×exp(-s)GN-1(t, s).
GN+1(t, s)=-(1+t)N+1LNtst+1-LN+1tst+1+GN(t, s)+exp(s)(1+t)N+1-1+tt [(1+t)N+1-(1+t)N]+(1+t)×exp(-s)[GN(t, s)-GN-1(t, s)]=-(1+t)N+1LNtst+1-LN+1tst+1+(2+t)GN(t, s)-(1+t)GN-1(t, s),

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