Abstract

Relations connecting the parameters of a super-Gaussian with those of a flattened Gaussian beam are determined by minimizing the mean squared difference of the two profiles. Simplified analytical expressions are suggested and tested for values of the power parameter of the super-Gaussian function up to 20.

© 1999 Optical Society of America

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References

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  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]
  2. A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
    [CrossRef]
  3. B. Lü, B. Zhang, X. Wang, “Three-dimensional intensity distribution of focused super-Gaussian beams,” Opt. Commun. 126, 1–6 (1996).
    [CrossRef]
  4. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  5. C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
    [CrossRef]
  6. S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
    [CrossRef]
  7. 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]
  8. M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
    [CrossRef]
  9. R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
    [CrossRef]
  10. S.-A. Amarande, “Beam propagation factor and the Kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
    [CrossRef]
  11. C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
    [CrossRef]
  12. 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]

1998 (1)

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

1997 (2)

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

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

1996 (4)

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]

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

1994 (2)

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

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (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]

Aiello, D.

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.

Bollanti, S.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Borghi, R.

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (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 of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

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]

Di Lazzaro, P.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Gori, F.

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]

Lü, B.

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

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]

Morin, M.

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

Murra, D.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Pacileo, A. M.

Palma, C.

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Parent, A.

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

Saghafi, S.

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

Santarsiero, M.

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (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 of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Sheppard, C. J. R.

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

Spagnolo, G. Schirripa

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]

Torre, A.

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Vicalvi, S.

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

Wang, X.

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

Zhang, B.

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

IEEE J. Quantum Electron. (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]

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. A (1)

Opt. Commun. (7)

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

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

C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

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

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

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

S. Bollanti, P. Di Lazzaro, D. Murra, A. Torre, “Analytical propagation of supergaussian-like beams in the far-field,” Opt. Commun. 138, 35–39 (1997).
[CrossRef]

Opt. Quantum Electron. (1)

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

Other (1)

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]

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Figures (5)

Fig. 1
Fig. 1

Ratio of the width parameters of corresponding FG and SG profiles as a function of γ.

Fig. 2
Fig. 2

Order N of a FG profile as a function of the parameter γ of the corresponding SG profile.

Fig. 3
Fig. 3

Error function, defined in Eq. (3), between corresponding SG and FG profiles as a function of γ. Parameters of the FG profile are evaluated by means of Eqs. (4) and (5).

Fig. 4
Fig. 4

Comparison between SG (solid curves) and FG (dotted curves) profiles for (a) wSG=1, γ=4, (b) γ=6, and (c) γ=8. Parameters of the FG curve are evaluated by means of Eqs. (4) and (5).

Fig. 5
Fig. 5

Difference between SG and FG profiles for the three cases of Fig. 4: γ=4 (solid curve), γ=6 (dashed curve), γ=8 (dotted curve).

Equations (5)

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SG(x; wSG, γ)=A exp-|x|wSGγ,
FG(x; wFG, N)=A exp-(N+1)x2wFG2×n=0N 1n!(N+1)x2wFG2n,
ϵ=-|SG(x; wSG, γ)-FG(x; wFG, N)|2dx1/2.
wFGwSG=1ifγ2.40.95if2.4<γ5,0.916γ1/44ifγ>5
N=0ifγ2.4int{0.18γ2}ifγ>2.4,

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