Abstract

The closed-form propagation equation of flattened Gaussian beams passing through a paraxial optical ABCD system, in which the linear gain and absorption media are included, is derived, and its general applicable advantage is illustrated with numerical examples.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  2. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  3. S.-A. Amarande, “Beam propagation factor and kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
    [CrossRef]
  4. M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focus-ing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
    [CrossRef]
  5. R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
    [CrossRef]
  6. B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–287 (1999).
  7. 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]
  8. B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 20, 4581–4584 (1999).
  9. R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
    [CrossRef]
  10. B. Lü, S. Luo, B. Zhang, “Propagation of three-dimensional flattened Gaussian beams,” J. Mod. Opt. 46, 1753–1762 (1999).
    [CrossRef]
  11. C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
    [CrossRef]
  12. C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
    [CrossRef]
  13. S. A. Collins, “Lens-system diffraction integral written terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  14. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, pp. 146, 386.

1999 (4)

B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–287 (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, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 20, 4581–4584 (1999).

B. Lü, S. Luo, B. Zhang, “Propagation of three-dimensional flattened Gaussian beams,” J. Mod. Opt. 46, 1753–1762 (1999).
[CrossRef]

1998 (2)

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 partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

1997 (1)

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

1996 (3)

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

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

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[CrossRef]

1995 (1)

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

1994 (1)

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

1970 (1)

Aiello, D.

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

Amarande, S.-A.

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

Bagini, V.

Borghi, R.

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 partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focus-ing 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, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Cincotti, G.

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[CrossRef]

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

Collins, S. A.

De Santis, P.

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[CrossRef]

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, pp. 146, 386.

Gori, F.

Guattari, G.

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[CrossRef]

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

Lü, B.

B. Lü, S. Luo, B. Zhang, “Propagation of three-dimensional flattened Gaussian beams,” J. Mod. Opt. 46, 1753–1762 (1999).
[CrossRef]

B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–287 (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, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 20, 4581–4584 (1999).

Luo, S.

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, 285–287 (1999).

B. Lü, S. Luo, B. Zhang, “Propagation of three-dimensional flattened Gaussian beams,” J. Mod. Opt. 46, 1753–1762 (1999).
[CrossRef]

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 20, 4581–4584 (1999).

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, pp. 146, 386.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, pp. 146, 386.

Pacileo, A. M.

Palma, C.

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[CrossRef]

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

Santarsiero, M.

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

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

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, pp. 146, 386.

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, “Focus-ing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

Zhang, B.

B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–287 (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, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 20, 4581–4584 (1999).

B. Lü, S. Luo, B. Zhang, “Propagation of three-dimensional flattened Gaussian beams,” J. Mod. Opt. 46, 1753–1762 (1999).
[CrossRef]

Appl. Opt. (1)

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 20, 4581–4584 (1999).

J. Mod. Opt. (4)

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

B. Lü, S. Luo, B. Zhang, “Propagation of three-dimensional flattened Gaussian beams,” J. Mod. Opt. 46, 1753–1762 (1999).
[CrossRef]

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation of partially coherent beams in absorbing media,” J. Mod. Opt. 42, 1123–1135 (1995).
[CrossRef]

C. Palma, P. De Santis, G. Cincotti, G. Guattari, “Propagation and coherence evolution of optical beams in gain media,” J. Mod. Opt. 43, 139–153 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (4)

S.-A. Amarande, “Beam propagation factor and 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]

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

Opt. Lett. (1)

Optik (1)

B. Lü, S. Luo, B. Zhang, “A comparison between the flattened Gaussian beam and super-Gaussian beam,” Optik 110, 285–287 (1999).

Other (1)

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1, pp. 146, 386.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Relative intensity distributions of a FGB propagating in free space (solid curves), linear gain medium (dashed curves), and linear absorption medium (dotted curves). (a) l=0, (b) l=530 mm, (c) l=5 m. The other calculation parameters are N=10, w0=1 mm, |k|=5000/mm, and η=±10-5 for the gain and the absorption media, respectively.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

E(x, 0)=A0exp-N+1w02 x2n=0N1n!N+1w02 x2n,
E(x, z)=ik2πB1/2exp-ikz2-E(x, 0)×exp-ik2B (Ax2-2xx+Dx2)dx,
k=kr+iki,
E(x, z)
=A0ikr-ki2B1/2exp-ikr-ki2 zexp-(ikr-ki)D2B x2N+1w02+(ikr-ki)A2B1/2×exp-(kr+iki)xB24N+1w02+(ikr-ki)A2B×n=0N-14n1n!N+1w02nN+1w02+(ikr-ki)A2B-n×H2n(kr+iki)xB2N+1w02+(ikr-ki)A2B1/2,
E(x, z)=A0ik2B1/2exp-ikz2exp-ikD2B x2N+1w02+ikA2B1/2×exp-(kx/B)24N+1w02+ikA2B×n=0N-14n1n!N+1w02nN+1w02+ikA2B-n×H2nkx/B2N+1w02+ikA2B1/2,
E(x, l)
=A0ikr-ki2l1/2exp-ikr-ki2 lexp-ikr-ki2l x2N+1w02+ikr-ki2l1/2×exp-(kr+iki)2x2l24N+1w02+ikr-ki2l×n=0N-14n1n!N+1w02nN+1w02+ikr-ki2l-n×H2n(kr+iki)xl2N+1w02+ikr-ki2l1/2.
lw02|k|2η(N+1),
η=ki/|k|.
AiBiCiDi(i=1, 2),
0tν-1exp-t28aexp(-pt)dt
=Γ(ν)2νa(1/2)νexp(ap2)D-ν(2pa1/2),
Re a>0, Reν>0,
E(x, z)=ik2πB1/2exp-ikz2exp-ikDx22B×exp-k2x28B2N+1w02+ikA2B×n=0N1n! Γ(2n+1)2-n-1/2N+1w02n×N+1w02+ikA2B-n-1/2S,
S=D-(2n+1)-ikxB2N+1w02+ikA2B1/2+D-(2n+1)ikxB2N+1w02+ikA2B1/2.
Dn(z)=2n/2+1/4z-1/2Wn/2+1/4,1/4(z2/2),
Wk,u(z)=Γ(-2u)Mk,u(z)Γ(12-u-k)+Γ(2u)Mk,-u(z)Γ(12+u-k),
Mk,u(z)=21/2+uexp(-z/2)×F(12+u-k; 2u+1; z).
S=Γ(12)2-n+1/2Γ(n+1)expk2x28B2N+1w02+ikA2B×Fn+12, 12, -k2x24B2N+1w02+ikA2B .
F(α, γ, z)=exp(z)F(γ-α, γ, -z),
H2m(x)=(-1)mm! (2m)!F-m; 12; x2,

Metrics