Abstract

A simple model called partially coherent flattened Gaussian beam (FGB) is proposed to describe a partially coherent beam with a flat-topped spatial profile. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent FGB propagating through a paraxial ABCD optical system. The propagation factor and propagation properties of a partially coherent FGB in free space are studied in detail and found to be closely related to its coherence and beam order.

© 2006 Optical Society of America

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References

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  1. M. S. Bowers, "Diffractive analysis of unstable optical resonator with super-Gaussian mirrors," Opt. Lett. 19, 1319-1321 (1992).
    [CrossRef]
  2. F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
    [CrossRef]
  3. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa, "Propagation 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).
  5. R. Borghi, M. Santarsiero, and S. Vicalvi, "Focal shift of focused flat-topped light beams," Opt. Commun. 154, 243-248 (1998).
    [CrossRef]
  6. Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
    [CrossRef]
  7. M. Ibnchaikh and A. Belafhal, "Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system," Opt. Commun. 193, 73-79 (2001).
    [CrossRef]
  8. Y. Li, "Light beam with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]
  9. Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
    [CrossRef]
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  11. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
    [CrossRef]
  12. A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
    [CrossRef]
  13. R. Borghi and M. Santarsiero, "Modal decomposition of partially coherent flat-topped beams produced by multimode lasers," Opt. Lett. 23, 313-315 (1998).
    [CrossRef]
  14. R. Borghi and M. Santarsiero, "Modal structure analysis for a class of axially symmetric flat-topped beams," IEEE J. Quantum Electron. 35, 745-750 (1999).
    [CrossRef]
  15. D. W. Coutts, "Time-resolved beam divergence from a copper-vapor laser with unstable resonator," IEEE J. Quantum Electron. 31, 330-342 (1995).
    [CrossRef]
  16. D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: Generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002).
    [CrossRef]
  17. J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).
  18. Y. G. Basov, "Divergence of excimer laser beams: a review," J. Commun. Technol. Electron. 46, 1-6 (2001).
  19. D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
    [CrossRef] [PubMed]
  20. G. Wu, H. Guo, and D. Deng, "Comment on 'Partially coherent flat-topped beam and its propagation'," Appl. Opt. 45, 366-368 (2006).
    [CrossRef] [PubMed]
  21. D. Xu, Y. Cai, D. Ge, and Q. Lin, "Reply to comment on 'Partially coherent flat-topped beam and its propagation'," Appl. Opt. 45, 369-371 (2006).
    [CrossRef]
  22. J. T. Foley and M. S. Zubairy, "Directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
    [CrossRef]
  23. A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2C14 (1990).
  24. F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
    [CrossRef]
  25. S. A. Collins, "Lens-system diffraction integral written terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970).
    [CrossRef]
  26. A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).
  27. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1970).

2006 (2)

2004 (2)

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

2003 (1)

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

2002 (2)

D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: Generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002).
[CrossRef]

Y. Li, "Light beam with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

2001 (2)

Y. G. Basov, "Divergence of excimer laser beams: a review," J. Commun. Technol. Electron. 46, 1-6 (2001).

M. Ibnchaikh and A. Belafhal, "Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system," Opt. Commun. 193, 73-79 (2001).
[CrossRef]

2000 (1)

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

1999 (1)

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

1998 (2)

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

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

1996 (2)

1995 (1)

D. W. Coutts, "Time-resolved beam divergence from a copper-vapor laser with unstable resonator," IEEE J. Quantum Electron. 31, 330-342 (1995).
[CrossRef]

1994 (1)

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1993 (1)

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

1992 (1)

1991 (1)

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

1978 (1)

J. T. Foley and M. S. Zubairy, "Directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

1970 (1)

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1970).

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.

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Bagini, V.

Basov, Y. G.

Y. G. Basov, "Divergence of excimer laser beams: a review," J. Commun. Technol. Electron. 46, 1-6 (2001).

Belafhal, A.

M. Ibnchaikh and A. Belafhal, "Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system," Opt. Commun. 193, 73-79 (2001).
[CrossRef]

Belendez, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Borghi, R.

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

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

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

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

Bowers, M. S.

Cai, Y.

D. Xu, Y. Cai, D. Ge, and Q. Lin, "Reply to comment on 'Partially coherent flat-topped beam and its propagation'," Appl. Opt. 45, 369-371 (2006).
[CrossRef]

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Carretero, L.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Collins, S. A.

Coutts, D. W.

D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: Generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002).
[CrossRef]

D. W. Coutts, "Time-resolved beam divergence from a copper-vapor laser with unstable resonator," IEEE J. Quantum Electron. 31, 330-342 (1995).
[CrossRef]

Deng, D.

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Fimia, A.

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

Foley, J. T.

J. T. Foley and M. S. Zubairy, "Directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

Ge, D.

Gori, F.

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa, "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]

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Guo, H.

Ibnchaikh, M.

M. Ibnchaikh and A. Belafhal, "Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system," Opt. Commun. 193, 73-79 (2001).
[CrossRef]

Kaivola, M.

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Kajava, T.

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Kuittinen, M.

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Laakkonen, P.

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Li, Y.

Lin, Q.

D. Xu, Y. Cai, D. Ge, and Q. Lin, "Reply to comment on 'Partially coherent flat-topped beam and its propagation'," Appl. Opt. 45, 369-371 (2006).
[CrossRef]

D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Mandel, L.

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

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Paallonen, P.

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Pacileo, A. M.

Santarsiero, M.

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

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

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

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

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Schirripa, G.

Siegman, A. E.

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

Simonen, J.

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1970).

Turunen, J.

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Vicalvi, S.

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

Wolf, E.

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

Wu, G.

Xu, D.

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Zubairy, M. S.

J. T. Foley and M. S. Zubairy, "Directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (3)

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

D. W. Coutts, "Time-resolved beam divergence from a copper-vapor laser with unstable resonator," IEEE J. Quantum Electron. 31, 330-342 (1995).
[CrossRef]

D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: Generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002).
[CrossRef]

J. Commun. Technol. Electron. (1)

Y. G. Basov, "Divergence of excimer laser beams: a review," J. Commun. Technol. Electron. 46, 1-6 (2001).

J. Mod. Opt. (1)

J. Turunen, P. Paallonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

J. Opt. A, Pure Appl. Opt. (2)

Y. Cai and Q. Lin, "Properties of a flattened Gaussian beam in the fractional Fourier transform plane," J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (7)

J. T. Foley and M. S. Zubairy, "Directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297-300 (1978).
[CrossRef]

A. Belendez, L. Carretero, and A. Fimia, "The use of partially coherent light to reduce the efficiency of silver-halide noise gratings," Opt. Commun. 98, 236-240 (1993).
[CrossRef]

M. Ibnchaikh and A. Belafhal, "Closed-term propagation expressions of flattened Gaussian beams through an apertured ABCD optical system," Opt. Commun. 193, 73-79 (2001).
[CrossRef]

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

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

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, "Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression," Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Other (4)

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

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, 1970).

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2C14 (1990).

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Figures (4)

Fig. 1
Fig. 1

M 2 factor of a partially coherent FGB versus N, σ g , and w 0 .

Fig. 2
Fig. 2

Normalized irradiance distribution of a partially coherent FGB (with different coherence widths) as it propagates along the z direction in free space.

Fig. 3
Fig. 3

Normalized on-axis irradiance distribution of a partially coherent FGB (with several different beam orders) propagating along the z direction.

Fig. 4
Fig. 4

Modulus of the spectral degree of coherence of a partially coherent FGB (with different beam orders) for symmetric points ( u 2 = u 1 ) as a funtion of u 1 at several propagation distances in free space for order (a) 0, (b) 3.

Equations (22)

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

W ( x 1 , x 2 ; 0 ) = I ( x 1 ; 0 ) I ( x 2 ; 0 ) g ( x 1 x 2 ) ,
g ( x 1 x 2 ) = exp [ ( x 1 x 2 ) 2 2 σ g 2 ] ,
I ( x i ; 0 ) = { exp [ ( N + 1 ) x i 2 w 0 2 ] n = 0 N 1 n ! [ ( N + 1 ) x i 2 w 0 2 ] n } 2 = [ exp ( x i 2 w 0 N 2 ) n = 1 N 1 n ! ( x i 2 w 0 N 2 ) n ] 2 , i = 1 , 2 ,
W ( x 1 , x 2 ; 0 ) = n = 0 N m = 0 N 1 n ! m ! ( x 1 2 w 0 N 2 ) n ( x 2 2 w 0 N 2 ) n exp [ x 1 2 + x 2 2 w 0 N 2 ( x 1 x 2 ) 2 2 σ g 2 ] ,
M 2 = 4 π Δ x Δ p ,
Δ x = 1 I ( x x ¯ ) 2 W ( x , x ; 0 ) d x ,
Δ p = 1 I ( p p ¯ ) 2 W ̃ ( p , p ; 0 ) d p ,
x ¯ = 1 I x W ( x , x ; 0 ) d x ,
p ¯ = 1 I p W ̃ ( p , p ) d p ,
I = W ( x , x ; 0 ) d x = W ̃ ( p , p ; 0 ) d p ,
M 2 = 2 [ W ( x , x ; 0 ) d x ] 1 { x 2 W ( x , x ; 0 ) d x [ 2 W ( x 1 , x 2 ; 0 ) x 1 x 2 ] x , x d x } 1 2 .
M 2 = 2 [ n = 0 N m = 0 N 1 n ! m ! 1 w 0 n 2 ( n + m ) ( 2 w 0 n 2 ) ( n + m + 3 2 ) Γ ( m + n + 3 2 ) ] 1 2 n = 0 N m = 0 N 1 n ! m ! 1 w 0 n 2 ( n + m ) ( 2 w 0 n 2 ) ( n + m + 1 2 ) Γ ( m + n + 1 2 ) × ( n = 0 N m = 0 N 2 n m n ! m ! { [ 1 σ g 2 4 ( m + n ) w 0 n 2 ] w 0 n 2 1 2 Γ ( m + n + 1 2 ) + m n w 0 n 2 5 2 Γ ( m + n 1 2 ) + 2 1 2 w 0 n Γ ( m + n + 3 2 ) } ) 1 2
W o ( u 1 , u 2 ; z ) = 1 λ b W ( x 1 , x 2 ; 0 ) exp [ i k 2 b ( a x 1 2 2 x 1 u 1 + d u 1 2 ) + i k 2 b ( a x 2 2 2 x 2 u 2 + d u 2 2 ) ] d x 1 d x 2 ,
W o ( u 1 , u 2 ; z ) = π λ b n = 0 N m = 0 N 1 n ! m ! ( 2 i ) 2 m w 0 N 2 m C 1 m + 1 2 exp ( i k d 2 b u 1 2 + i k d 2 b u 2 2 ) exp ( x 1 2 w 0 N 2 ) ( x 1 2 w 0 N 2 ) n exp ( x 1 2 2 σ g 2 ) exp [ i k 2 b ( a x 1 2 2 x 1 u 1 ) ] exp [ 1 4 C 1 ( x 1 σ g 2 i k b u 2 ) 2 ] H 2 m [ i 2 C 1 ( x 1 σ g 2 i k b u 2 ) ] d x 1 ,
C 1 = ( 1 w 0 N 2 + 1 2 σ g 2 i k a 2 b ) .
x n exp [ ( x β ) 2 ] d x = ( 2 i ) n π H n ( i β ) ,
H n ( x + y ) = 1 2 n 2 k = 0 n ( n k ) H k ( 2 x ) H n k ( 2 y ) ,
H n ( x ) = n ! m = 0 [ n 2 ] ( 1 ) m 1 m ! ( n 2 m ) ! ( 2 x ) n 2 m ,
W o ( u 1 , u 2 ; z ) = π λ b n = 0 N m = 0 N k = 0 2 m p = 0 [ h 2 ] ( 2 m h ) ( 1 ) p p ! ( h 2 p ) ! h ! n ! m ! ( 2 i ) 2 m w 0 N 2 m C 1 m + 1 2 1 2 m exp ( i k d 2 b u 1 2 + i k d 2 b u 2 2 k 2 u 2 2 4 C 1 b 2 ) H 2 m h ( k u 2 b 2 C 1 ) ( x 1 2 w 0 N 2 ) n ( i 2 x 1 C 1 σ g 2 ) h 2 p exp ( C 1 x 1 2 ) exp ( i k b x 1 u 1 i k x 1 u 2 2 C 1 σ g 2 b ) d x 1 ,
C 2 = ( 1 w 0 N 2 + 1 2 σ g 2 + i k a 2 b 1 4 C 1 σ g 4 ) .
W o ( u 1 , u 2 ; z ) = π λ b n = 0 N m = 0 N k = 0 2 m p = 0 [ h 2 ] ( 2 m h ) ( 1 ) p p ! ( h 2 p ) ! h ! n ! m ! 2 p 2 n 3 m h 2 w 0 N 2 m w 0 N 2 n C 1 m p + h 2 + 1 2 ( i ) 2 n 2 m C 2 n p + h 2 + 1 2 ( 1 σ g 2 ) h 2 p exp [ i k d 2 b u 1 2 + i k d 2 b u 2 2 k 2 u 2 2 4 C 1 b 2 ] exp [ k 2 4 C 2 b 2 ( u 1 u 2 2 C 1 σ g 2 ) 2 ] H 2 m h ( k u 2 b 2 C 1 ) H 2 n + h 2 p [ 1 2 C 2 ( k u 2 2 C 1 σ g 2 b k b u 1 ) ] .
g ( u 1 , u 2 ) = W ( u 1 , u 2 , z ) W ( u 1 , u 1 , z ) W ( u 1 , u 2 , z ) .

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