Abstract

Explicit expressions for the M 2 factor and field distribution of flattened Gaussian beams that propagate through an unapertured paraxial ABCD system are derived and illustrated with numerical examples. We show that the far-field expression of flattened Gaussian beams can be readily obtained by use of our results.

© 1999 Optical Society of America

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References

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  1. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  2. A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
    [CrossRef]
  3. C. Palma, V. Bagini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
    [CrossRef]
  4. C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
    [CrossRef]
  5. C. Palma, V. Bagini, “Expansions of general beams in Gaussian beams,” Opt. Commun. 116, 1–7 (1995).
    [CrossRef]
  6. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  7. M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Intensity and phase distribution of focused flattened Gaussian beams,” in Third International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds., Proc. SPIE2870, 288–296 (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. S.-A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
  10. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  11. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).
  12. I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).
  13. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1068–1177 (1970).
    [CrossRef]

1997

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

1996

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

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

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

1995

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

1994

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

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

1970

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]

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Intensity and phase distribution of focused flattened Gaussian beams,” in Third International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds., Proc. SPIE2870, 288–296 (1996).
[CrossRef]

Amarande, S.-A.

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

Ambrosini, D.

Bagini, V.

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

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

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

Borghi, R.

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. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Intensity and phase distribution of focused flattened Gaussian beams,” in Third International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds., Proc. SPIE2870, 288–296 (1996).
[CrossRef]

Collins, S. A.

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Gori, F.

Gradshtein, I. S.

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

Lavigne, P.

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

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Morin, M.

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

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Pacileo, A. M.

Palma, C.

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

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]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Tables 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.

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. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Intensity and phase distribution of focused flattened Gaussian beams,” in Third International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds., Proc. SPIE2870, 288–296 (1996).
[CrossRef]

Sheppard, C. J. R.

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

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Spagnolo, G. S.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Vicalvi, S.

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

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Intensity and phase distribution of focused flattened Gaussian beams,” in Third International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds., Proc. SPIE2870, 288–296 (1996).
[CrossRef]

J. Mod. Opt.

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.

J. Opt. Soc. Am. A

Opt. Commun.

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

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]

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

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

Opt. Quantum Electron.

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

Other

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

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

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Intensity and phase distribution of focused flattened Gaussian beams,” in Third International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds., Proc. SPIE2870, 288–296 (1996).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

M 2 factor as a function of beam order N for a FGB.

Fig. 2
Fig. 2

Relative intensity distribution [I(x, z)]/I(0, z)] of a FGB that propagates in free space. The calculated parameters are w 0 = 1 mm and λ = 1.06 µm at (a) z = 0, (b) z = 1 m, (c) z = 2.5 m, (d) z = 100 m.

Equations (16)

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Ex=exp-x2/w02m=0Nx/w02mm!,
Ix=ExE*x=exp-2x2w02m=0Nm=0Nx/w02m+mm!m!,
σx2=-+ x2Ixdx-+ Ixdx.
0+ exp-sttνdt=ν!s-ν-1
σx2=w022m=0Nm=0NΓm+m+3/22m+mm!m!m=0Nm=0NΓm+m+1/22m+mm!m!,
0+ x2m exp-α2x2cosxydx=-1mπ 2-m+1α-2m+1 exp-y24α2 He2my2 α,
ESx=exp-πw0Sx2m=0N-1/2mm!×He2m2 πw0Sx,
σk2=-+ Sx2IˆSxdSx-+ IˆSxdSx,
σk2=2w02m=0Nm=0Nj=0mj=0m-1/2m+m+j+j2m!2m!Γm+m-j-j+3/2m!m!j!j!2m-2j!2m-2j!m=0Nm=0NΓm+m+1/22m+mm!m!.
-+ exp-2x2HmxHnxdx=-1m+n/22m+n-1/2×Γ m+n+12m+n=even number
M2=2σxσk=m=0Nm=0NΓm+m+3/22m+mm!m!1/2m=0Nm=0Nj=0mj=0m-1/2m+m+j+j2m!2m!Γm+m-j-j+3/2m!m!j!j!2m-2j!2m-2j!1/2m=0Nm=0NΓm+m+1/22m+mm!m!.
Ex=-iλB-+ Ex×expik2BAx2-2xx+Dx2dx,
Ex=c exp-kxB241w02-ikA2Bm=0N-0.5mw02mm!×1w02-ikA2B-m He2mkxB21w02-ikA2B1/2,
c=-iπλB1w02-ikA2B-1/2 expikDx22B.
Ex, z=c exp-kx2z21w02-ik2zm=0N-0.5mw02mm!×1w02-ik2z-mHe2mkxz21w02-ik2z1/2.
Ix, z=Ex, zE*x, z,

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