Abstract

By means of the Huygens–Fresnel diffraction integral, the field representation of a laser beam modulated by a hard-edged aperture is derived. The near-field and far-field transverse intensity distributions of the beams with different bandwidths are analyzed by using the representation. The numerical calculation results indicate that the amplitudes and numbers of the intensity spikes decrease with increasing bandwidth, and beam smoothing is achieved when the bandwidth takes a certain value in the near field. In the far field, the radius of the transverse intensity distribution decreases as the bandwidth increases, and the physical explanation of this fact is also given. © 2005 Optical Society of America

© 2005 Optical Society of America

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References

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  1. P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [CrossRef]
  2. K. A. Bruecker, S. Jorna, “Laser-driven fusion,” Rev. Mod. Phys. 46, 325–367 (1974).
    [CrossRef]
  3. S. Skupsky, K. Lee, “Uniformity of energy deposition for laser driven fusion,” J. Appl. Phys. 54, 3662–3671 (1983).
    [CrossRef]
  4. V. R. Cositich, B. C. Johnson, “Apertures to shape highpower beams,” Laser Focus (Newton, Mass.) 10, 43–46 (1974).
  5. J. T. Hunt, P. A. Renard, W. W. Simmons, “Improved performance of fusion lasers using the imaging properties of multiple spatial filters,” Appl. Opt. 16, 779–782 (1977).
    [PubMed]
  6. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [CrossRef]
  7. X. M. Deng, X. C. Liang, Z. Z. Chen, W. Y. Yu, R. Y. Ma, “Uniform illumination of large targets using a lens array,” Appl. Opt. 25, 377–381 (1986).
    [CrossRef] [PubMed]
  8. S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
    [CrossRef]
  9. “Two-dimensional SSD on OMEGA,” in Laboratory for Laser Energetics Review (University of Rochester, 1996), Vol. 69, pp. 1–10.
  10. H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
    [CrossRef]
  11. J. E. Rothenberg, “Comparison of beam-smoothing methods for direct-drive inertial confinement fusion,” J. Opt. Soc. Am. B 14, 1664–1671 (1997).
    [CrossRef]
  12. H. A. Rose, S. Ghosal, “Effect of smoothing by spectral dispersion on flow induced laser beam deflection: the random phase modulation scheme,” Phys. Plasmas 5, 775–781 (1998).
    [CrossRef]
  13. G. Miyaji, N. Miyanaga, S. Urushihara, K. Suzuki, S. Matsuoka, M. Nakatsuka, A. Morimoto, T. Kobayashi, “Three-directional spectral dispersion for smoothing of a laser irradiance profile,” Opt. Lett. 27, 725–727 (2002).
    [CrossRef]
  14. Q. F. Tan, Y. B. Yan, G. F. Jin, “Statistic analysis of influence of phase distortion on diffractive optical element for beam smoothing,” Opt. Express 12, 3270–3278 (2004).
    [CrossRef] [PubMed]
  15. B. Schenkel, J. Biegert, U. Keller, “Generation of 3.8-fs pulses from adaptive compression of a cashed hollow fiber supercontinuum,” Opt. Lett. 28, 1987–1989 (2003).
    [CrossRef] [PubMed]
  16. I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
    [CrossRef]
  17. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
    [CrossRef]
  18. S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
    [CrossRef]
  19. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606-1–11 (2002).
    [CrossRef]
  20. J. J. Thomson, “Finite-bandwidth effects on the parametric instability in an inhomogeneous plasma,” Nucl. Fusion 15, 237–247 (1975).
    [CrossRef]
  21. K. Estabrook, W. L. Kruer, “Theory and simulation of one-dimensional Raman backward and forward scattering,” Phys. Fluids 26, 1892–1903 (1983).
    [CrossRef]
  22. A. E. Siegman, Lasers (University Science Books, 1986), Sect. 9.1.
  23. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.
    [CrossRef]
  24. R. W. Peng, D. Y. Fan, “Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam,” Opt. Commun. 246, 241–248 (2005).
    [CrossRef]

2005 (1)

R. W. Peng, D. Y. Fan, “Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam,” Opt. Commun. 246, 241–248 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

2000 (1)

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

1999 (1)

1998 (1)

H. A. Rose, S. Ghosal, “Effect of smoothing by spectral dispersion on flow induced laser beam deflection: the random phase modulation scheme,” Phys. Plasmas 5, 775–781 (1998).
[CrossRef]

1997 (1)

1993 (1)

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

1989 (1)

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

1986 (1)

1985 (1)

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

1984 (1)

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

1983 (2)

S. Skupsky, K. Lee, “Uniformity of energy deposition for laser driven fusion,” J. Appl. Phys. 54, 3662–3671 (1983).
[CrossRef]

K. Estabrook, W. L. Kruer, “Theory and simulation of one-dimensional Raman backward and forward scattering,” Phys. Fluids 26, 1892–1903 (1983).
[CrossRef]

1977 (1)

1975 (1)

J. J. Thomson, “Finite-bandwidth effects on the parametric instability in an inhomogeneous plasma,” Nucl. Fusion 15, 237–247 (1975).
[CrossRef]

1974 (2)

V. R. Cositich, B. C. Johnson, “Apertures to shape highpower beams,” Laser Focus (Newton, Mass.) 10, 43–46 (1974).

K. A. Bruecker, S. Jorna, “Laser-driven fusion,” Rev. Mod. Phys. 46, 325–367 (1974).
[CrossRef]

1965 (1)

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Arinaga, S.

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

Azechi, H.

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

Biegert, J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.
[CrossRef]

Bruecker, K. A.

K. A. Bruecker, S. Jorna, “Laser-driven fusion,” Rev. Mod. Phys. 46, 325–367 (1974).
[CrossRef]

Chen, Z. Z.

Christov, I. P.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Cositich, V. R.

V. R. Cositich, B. C. Johnson, “Apertures to shape highpower beams,” Laser Focus (Newton, Mass.) 10, 43–46 (1974).

Craxton, R. S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Deng, X. M.

Estabrook, K.

K. Estabrook, W. L. Kruer, “Theory and simulation of one-dimensional Raman backward and forward scattering,” Phys. Fluids 26, 1892–1903 (1983).
[CrossRef]

Fan, D. Y.

R. W. Peng, D. Y. Fan, “Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam,” Opt. Commun. 246, 241–248 (2005).
[CrossRef]

Feng, S.

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

Ghosal, S.

H. A. Rose, S. Ghosal, “Effect of smoothing by spectral dispersion on flow induced laser beam deflection: the random phase modulation scheme,” Phys. Plasmas 5, 775–781 (1998).
[CrossRef]

Hunt, J. T.

Jin, G. F.

Jitsuno, T.

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

Johnson, B. C.

V. R. Cositich, B. C. Johnson, “Apertures to shape highpower beams,” Laser Focus (Newton, Mass.) 10, 43–46 (1974).

Jorna, S.

K. A. Bruecker, S. Jorna, “Laser-driven fusion,” Rev. Mod. Phys. 46, 325–367 (1974).
[CrossRef]

Kanabe, T.

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

Kato, Y.

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

Keller, U.

Kelly, P. L.

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Kessler, T.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Kitagawa, Y.

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

Kobayashi, T.

Kruer, W. L.

K. Estabrook, W. L. Kruer, “Theory and simulation of one-dimensional Raman backward and forward scattering,” Phys. Fluids 26, 1892–1903 (1983).
[CrossRef]

Lee, K.

S. Skupsky, K. Lee, “Uniformity of energy deposition for laser driven fusion,” J. Appl. Phys. 54, 3662–3671 (1983).
[CrossRef]

Letzring, S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Liang, X. C.

Ma, R. Y.

Matsuoka, S.

Mima, K.

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

Miyaji, G.

Miyanaga, N.

G. Miyaji, N. Miyanaga, S. Urushihara, K. Suzuki, S. Matsuoka, M. Nakatsuka, A. Morimoto, T. Kobayashi, “Three-directional spectral dispersion for smoothing of a laser irradiance profile,” Opt. Lett. 27, 725–727 (2002).
[CrossRef]

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

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

Morimoto, A.

Nakai, S.

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

Nakano, H.

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

Nakatsuka, M.

G. Miyaji, N. Miyanaga, S. Urushihara, K. Suzuki, S. Matsuoka, M. Nakatsuka, A. Morimoto, T. Kobayashi, “Three-directional spectral dispersion for smoothing of a laser irradiance profile,” Opt. Lett. 27, 725–727 (2002).
[CrossRef]

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

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

Peng, R. W.

R. W. Peng, D. Y. Fan, “Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam,” Opt. Commun. 246, 241–248 (2005).
[CrossRef]

Porras, M. A.

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606-1–11 (2002).
[CrossRef]

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
[CrossRef]

Renard, P. A.

Rose, H. A.

H. A. Rose, S. Ghosal, “Effect of smoothing by spectral dispersion on flow induced laser beam deflection: the random phase modulation scheme,” Phys. Plasmas 5, 775–781 (1998).
[CrossRef]

Rothenberg, J. E.

Schenkel, B.

Short, R. W.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), Sect. 9.1.

Simmons, W. W.

Skupsky, S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

S. Skupsky, K. Lee, “Uniformity of energy deposition for laser driven fusion,” J. Appl. Phys. 54, 3662–3671 (1983).
[CrossRef]

Soures, J. M.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

Suzuki, K.

Tan, Q. F.

Thomson, J. J.

J. J. Thomson, “Finite-bandwidth effects on the parametric instability in an inhomogeneous plasma,” Nucl. Fusion 15, 237–247 (1975).
[CrossRef]

Tsubakimoto, K.

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

Urushihara, S.

Winful, H. G.

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.
[CrossRef]

Yamanaka, C.

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

Yan, Y. B.

Yu, W. Y.

Appl. Opt. (2)

J. Appl. Phys. (3)

S. Skupsky, K. Lee, “Uniformity of energy deposition for laser driven fusion,” J. Appl. Phys. 54, 3662–3671 (1983).
[CrossRef]

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, J. M. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” J. Appl. Phys. 66, 3456–3462 (1989).
[CrossRef]

H. Nakano, K. Tsubakimoto, N. Miyanaga, M. Nakatsuka, T. Kanabe, H. Azechi, T. Jitsuno, S. Nakai, “Spectrally dispersed amplified spontaneous emission for improving irradiation uniformity into high power Nd:glass laser system,” J. Appl. Phys. 73, 2122–2130 (1993).
[CrossRef]

J. Opt. Soc. Am. B (2)

Laser Focus (Newton, Mass.) (1)

V. R. Cositich, B. C. Johnson, “Apertures to shape highpower beams,” Laser Focus (Newton, Mass.) 10, 43–46 (1974).

Nucl. Fusion (1)

J. J. Thomson, “Finite-bandwidth effects on the parametric instability in an inhomogeneous plasma,” Nucl. Fusion 15, 237–247 (1975).
[CrossRef]

Opt. Commun. (2)

R. W. Peng, D. Y. Fan, “Comparison between complex amplitude envelope representation and complex analytic signal representation in studying pulsed Gaussian beam,” Opt. Commun. 246, 241–248 (2005).
[CrossRef]

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Fluids (1)

K. Estabrook, W. L. Kruer, “Theory and simulation of one-dimensional Raman backward and forward scattering,” Phys. Fluids 26, 1892–1903 (1983).
[CrossRef]

Phys. Plasmas (1)

H. A. Rose, S. Ghosal, “Effect of smoothing by spectral dispersion on flow induced laser beam deflection: the random phase modulation scheme,” Phys. Plasmas 5, 775–781 (1998).
[CrossRef]

Phys. Rev. E (2)

S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E 61, 862–873 (2000).
[CrossRef]

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606-1–11 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

P. L. Kelly, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

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

Rev. Mod. Phys. (1)

K. A. Bruecker, S. Jorna, “Laser-driven fusion,” Rev. Mod. Phys. 46, 325–367 (1974).
[CrossRef]

Other (3)

“Two-dimensional SSD on OMEGA,” in Laboratory for Laser Energetics Review (University of Rochester, 1996), Vol. 69, pp. 1–10.

A. E. Siegman, Lasers (University Science Books, 1986), Sect. 9.1.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.2.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Transverse intensity distributions of beams with different bandwidths in the near field.

Fig. 2
Fig. 2

Time-averaged transverse intensity distributions of beams with different bandwidths in the near field.

Fig. 3
Fig. 3

(a) Transverse intensity distributions of beams with different bandwidths in the far field. (b) Time-averaged transverse intensity distributions of beams with different bandwidths in the far field.

Fig. 4
Fig. 4

Broadening factor of the transverse intensity distributions versus bandwidth.

Equations (21)

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E ( x , z , t ) = 1 ( 2 π ) 1 2 E ̃ ( x , z , ω ) exp ( i ω t ) d ω .
E ̃ ( x , z , ω ) = ( i λ z ) 1 2 exp ( i k z ) a a E ̃ 0 ( x 0 , 0 , ω ) exp [ i k 2 z ( x 0 x ) 2 ] d x 0 ,
E ̃ 0 ( x 0 , 0 , ω ) = 1 ( 2 π ) 1 2 E 0 ( x 0 , 0 , t ) exp ( i ω t ) d t
E 0 ( x 0 , 0 , t ) = E 0 ( x 0 , 0 ) f ( t ) ,
E ̃ 0 ( x 0 , 0 , ω ) = E 0 ( x 0 , 0 ) f ̃ ( ω ) ,
f ̃ ( ω ) = 1 ( 2 π ) 1 2 f ( t ) exp ( i ω t ) d t .
E 0 ( x 0 , 0 ) = exp ( i k 2 R x 0 2 ) ,
f ( t ) = exp ( a g 2 t 2 T P 2 ) cos ( ω 0 t ) ,
f ̃ ( ω ) = a g 2 ω 0 γ exp [ a g 2 ( ω ω 0 ) 2 ω 0 2 γ 2 ] .
E ̃ ( x , z , ω ) = [ R 4 ( z + R ) ] 1 2 exp ( i k z ) exp [ i k x 2 2 ( z + R ) ] [ erf ( χ + ) erf ( χ ) ] f ̃ ( ω ) ,
χ + = [ i k ( z + R ) 2 z R ] 1 2 ( R x z + R + a ) ,
χ = [ i k ( z + R ) 2 z R ] 1 2 ( R x z + R a ) ,
erf ( y ) = 2 π 1 2 0 y exp ( x 2 ) d x
E ( x , z , t ) = [ R 8 π ( z + R ) ] 1 2 [ erf ( χ + ) erf ( χ ) ] f ̃ ( ω ) exp ( i ω τ ) d ω ,
τ = τ x 2 2 c ( z + R ) = t z c x 2 2 c ( z + R ) ,
I ( x , z , t ) = R 8 π ( z + R ) [ erf ( χ + ) erf ( χ ) ] f ̃ ( ω ) exp ( i ω τ ) d ω 2 .
E ( x , z , t ) = [ R 4 ( z + R ) ] 1 2 exp ( i ω 0 τ ω 0 2 γ 2 τ 2 4 a g ) ξ ,
ξ = 1 2 { exp ( 2 i ω 0 τ ) [ 1 erf ( a g γ i γ ω 0 τ 2 a g ) ] + [ 1 erf ( a g γ i γ ω 0 τ 2 a g ) ] } .
δ = exp ( ω 0 2 γ 2 τ 2 4 a g ) .
δ = exp [ ω 0 2 γ 2 x 4 16 a g c 2 ( z + R ) 2 ] .
x HM 2 [ ln 2 a g c 2 ( z + R ) 2 ω 0 2 γ 2 ] 1 4 .

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