Abstract

A proposal for spectral broadening of an intense laser pulse with energy 15 mJ and a pulse duration of 40 fs in an argon-filled hollow fiber, using conjugate pressure-gradient method, is presented. The design, to be referred as conjugate pressure-gradient method, shall consist of a pair of hollow-fiber segments, each with gas pressure that increasingly and decreasingly varies, respectively. It is expected that the occurrence of early self-focusing can be avoided, as well as the elimination of the additional nonlinear processes at hollow fiber exit. Simulation using 15 mJ, 40 fs laser pulses shows that, in the intensity regime where ionization is important, excellent spectral broadening and spectral phase of the pulses can be obtained using the proposed conjugate pressure gradient method. The method therefore shall be useful for the spectral broadening of high-energy, ultrashort laser pulses.

© 2005 Optical Society of America

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  1. T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
    [CrossRef]
  2. Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
    [CrossRef]
  3. A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
    [CrossRef] [PubMed]
  4. M. Nisoli, S. De Silvestri, and O. Svelto, "Generation of high energy 10 fs pulse by a new compression technique," Appl. Phys. Lett. 68, 2793-2795 (1996)
    [CrossRef]
  5. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, "Compression of high-energy laser pulses below 5 fs," Opt. Lett. 22, 522-524 (1997).
    [CrossRef] [PubMed]
  6. J.-C. Diels and W. Rudolf, Ultrashort Laser Phenomena (Academic, 1996).
  7. N. Karasawa, R. Morita, H. Shigekawa, and M. Yamashita, "Generation of intense ultra broadband optical pulses by induced phase modulation in an argon-filled single-mode hollow waveguide," Opt. Lett. 25, 83-85 (2000).
    [CrossRef]
  8. M. Nurhuda, A. Suda, and K. Midorikawa, "Ionization-induced high order nonlinear susceptibility," Phys. Rev. A 66, 041802(R) (2002).
    [CrossRef]
  9. M. Nurhuda, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Propagation dynamics of femtosecond laser pulses in a hollow fiber filled with argon: constant gas pressure versus differential gas pressure," J. Opt. Soc. Am. B 20, 2002-2011 (2003).
    [CrossRef]
  10. A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Generation of sub-10-fs, 5-mJ optical pulses using a hollow fiber with a pressure gradient," Appl. Phys. Lett. 86111116 (2005).
    [CrossRef]
  11. N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
    [CrossRef]
  12. A. M. Perelemov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (2004).
  13. S. F. J. Larochelle, A. Talebpour, and S. L. Chin, "Coulomb effect in multiphoton ionization of rare-gas atom," J. Phys. B 31, 1215-1224 (1988).
    [CrossRef]
  14. N. H. Shon, A. Suda, Y. Tamaki, and K. Midorikawa, "High-order harmonic and attosecond pulse generations: Bulk media versus hollow waveguides," Phys. Rev. A 63, 063806 (2001).
    [CrossRef]
  15. Catalog (CVI Laser corporation, Albuquerque, New Mexico, 1997).

2005

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Generation of sub-10-fs, 5-mJ optical pulses using a hollow fiber with a pressure gradient," Appl. Phys. Lett. 86111116 (2005).
[CrossRef]

2004

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

A. M. Perelemov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (2004).

2003

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

M. Nurhuda, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Propagation dynamics of femtosecond laser pulses in a hollow fiber filled with argon: constant gas pressure versus differential gas pressure," J. Opt. Soc. Am. B 20, 2002-2011 (2003).
[CrossRef]

2002

M. Nurhuda, A. Suda, and K. Midorikawa, "Ionization-induced high order nonlinear susceptibility," Phys. Rev. A 66, 041802(R) (2002).
[CrossRef]

2001

N. H. Shon, A. Suda, Y. Tamaki, and K. Midorikawa, "High-order harmonic and attosecond pulse generations: Bulk media versus hollow waveguides," Phys. Rev. A 63, 063806 (2001).
[CrossRef]

2000

1999

Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
[CrossRef]

1997

1996

M. Nisoli, S. De Silvestri, and O. Svelto, "Generation of high energy 10 fs pulse by a new compression technique," Appl. Phys. Lett. 68, 2793-2795 (1996)
[CrossRef]

1988

S. F. J. Larochelle, A. Talebpour, and S. L. Chin, "Coulomb effect in multiphoton ionization of rare-gas atom," J. Phys. B 31, 1215-1224 (1988).
[CrossRef]

Backus, S.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Bartels, R. A.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Brabec, T.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[CrossRef]

Chin, S. L.

S. F. J. Larochelle, A. Talebpour, and S. L. Chin, "Coulomb effect in multiphoton ionization of rare-gas atom," J. Phys. B 31, 1215-1224 (1988).
[CrossRef]

Christov, I. P.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

De Silvestri, S.

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, "Compression of high-energy laser pulses below 5 fs," Opt. Lett. 22, 522-524 (1997).
[CrossRef] [PubMed]

M. Nisoli, S. De Silvestri, and O. Svelto, "Generation of high energy 10 fs pulse by a new compression technique," Appl. Phys. Lett. 68, 2793-2795 (1996)
[CrossRef]

Diels, J.-C.

J.-C. Diels and W. Rudolf, Ultrashort Laser Phenomena (Academic, 1996).

Ferencz, K.

Gibson, E. A.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

Green, H.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Hatayama, M.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Generation of sub-10-fs, 5-mJ optical pulses using a hollow fiber with a pressure gradient," Appl. Phys. Lett. 86111116 (2005).
[CrossRef]

M. Nurhuda, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Propagation dynamics of femtosecond laser pulses in a hollow fiber filled with argon: constant gas pressure versus differential gas pressure," J. Opt. Soc. Am. B 20, 2002-2011 (2003).
[CrossRef]

Itatani, J.

Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
[CrossRef]

Kapteyn, H. C.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Karasawa, N.

Krausz, F.

Larochelle, S. F.

S. F. J. Larochelle, A. Talebpour, and S. L. Chin, "Coulomb effect in multiphoton ionization of rare-gas atom," J. Phys. B 31, 1215-1224 (1988).
[CrossRef]

Midorikawa, K.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Generation of sub-10-fs, 5-mJ optical pulses using a hollow fiber with a pressure gradient," Appl. Phys. Lett. 86111116 (2005).
[CrossRef]

M. Nurhuda, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Propagation dynamics of femtosecond laser pulses in a hollow fiber filled with argon: constant gas pressure versus differential gas pressure," J. Opt. Soc. Am. B 20, 2002-2011 (2003).
[CrossRef]

M. Nurhuda, A. Suda, and K. Midorikawa, "Ionization-induced high order nonlinear susceptibility," Phys. Rev. A 66, 041802(R) (2002).
[CrossRef]

N. H. Shon, A. Suda, Y. Tamaki, and K. Midorikawa, "High-order harmonic and attosecond pulse generations: Bulk media versus hollow waveguides," Phys. Rev. A 63, 063806 (2001).
[CrossRef]

Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
[CrossRef]

Morita, R.

Murname, M.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

Murnane, M. M.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Nagasaka, K.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Generation of sub-10-fs, 5-mJ optical pulses using a hollow fiber with a pressure gradient," Appl. Phys. Lett. 86111116 (2005).
[CrossRef]

M. Nurhuda, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Propagation dynamics of femtosecond laser pulses in a hollow fiber filled with argon: constant gas pressure versus differential gas pressure," J. Opt. Soc. Am. B 20, 2002-2011 (2003).
[CrossRef]

Nagata, Y.

Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
[CrossRef]

Nisoli, M.

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, "Compression of high-energy laser pulses below 5 fs," Opt. Lett. 22, 522-524 (1997).
[CrossRef] [PubMed]

M. Nisoli, S. De Silvestri, and O. Svelto, "Generation of high energy 10 fs pulse by a new compression technique," Appl. Phys. Lett. 68, 2793-2795 (1996)
[CrossRef]

Nurhuda, M.

Obara, M.

Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
[CrossRef]

Paul, A.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Perelemov, A. M.

A. M. Perelemov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (2004).

Popmintchev, T.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

Popov, V. S.

A. M. Perelemov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (2004).

Rudolf, W.

J.-C. Diels and W. Rudolf, Ultrashort Laser Phenomena (Academic, 1996).

Sartania, S.

Shigekawa, H.

Shon, N. H.

N. H. Shon, A. Suda, Y. Tamaki, and K. Midorikawa, "High-order harmonic and attosecond pulse generations: Bulk media versus hollow waveguides," Phys. Rev. A 63, 063806 (2001).
[CrossRef]

Spielmann, Ch.

Suda, A.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Generation of sub-10-fs, 5-mJ optical pulses using a hollow fiber with a pressure gradient," Appl. Phys. Lett. 86111116 (2005).
[CrossRef]

M. Nurhuda, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Propagation dynamics of femtosecond laser pulses in a hollow fiber filled with argon: constant gas pressure versus differential gas pressure," J. Opt. Soc. Am. B 20, 2002-2011 (2003).
[CrossRef]

M. Nurhuda, A. Suda, and K. Midorikawa, "Ionization-induced high order nonlinear susceptibility," Phys. Rev. A 66, 041802(R) (2002).
[CrossRef]

N. H. Shon, A. Suda, Y. Tamaki, and K. Midorikawa, "High-order harmonic and attosecond pulse generations: Bulk media versus hollow waveguides," Phys. Rev. A 63, 063806 (2001).
[CrossRef]

Svelto, O.

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, "Compression of high-energy laser pulses below 5 fs," Opt. Lett. 22, 522-524 (1997).
[CrossRef] [PubMed]

M. Nisoli, S. De Silvestri, and O. Svelto, "Generation of high energy 10 fs pulse by a new compression technique," Appl. Phys. Lett. 68, 2793-2795 (1996)
[CrossRef]

Szipocs, R.

Talebpour, A.

S. F. J. Larochelle, A. Talebpour, and S. L. Chin, "Coulomb effect in multiphoton ionization of rare-gas atom," J. Phys. B 31, 1215-1224 (1988).
[CrossRef]

Tamaki, Y.

N. H. Shon, A. Suda, Y. Tamaki, and K. Midorikawa, "High-order harmonic and attosecond pulse generations: Bulk media versus hollow waveguides," Phys. Rev. A 63, 063806 (2001).
[CrossRef]

Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
[CrossRef]

Terent'ev, M. V.

A. M. Perelemov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (2004).

Tobey, R.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Wagner, N. L.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

Weiman, S.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Yamashita, M.

Appl. Phys. Lett.

M. Nisoli, S. De Silvestri, and O. Svelto, "Generation of high energy 10 fs pulse by a new compression technique," Appl. Phys. Lett. 68, 2793-2795 (1996)
[CrossRef]

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, "Generation of sub-10-fs, 5-mJ optical pulses using a hollow fiber with a pressure gradient," Appl. Phys. Lett. 86111116 (2005).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. B

S. F. J. Larochelle, A. Talebpour, and S. L. Chin, "Coulomb effect in multiphoton ionization of rare-gas atom," J. Phys. B 31, 1215-1224 (1988).
[CrossRef]

Nature

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, "Quasi-phase-matched generation of coherent extreme-ultraviolet light," Nature 421, 51-54 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. A

N. H. Shon, A. Suda, Y. Tamaki, and K. Midorikawa, "High-order harmonic and attosecond pulse generations: Bulk media versus hollow waveguides," Phys. Rev. A 63, 063806 (2001).
[CrossRef]

M. Nurhuda, A. Suda, and K. Midorikawa, "Ionization-induced high order nonlinear susceptibility," Phys. Rev. A 66, 041802(R) (2002).
[CrossRef]

Phys. Rev. Lett.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, S. Backus, M. Murname, and H. C. Kapteyn, "Self compression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004).
[CrossRef]

Y. Tamaki, J. Itatani, Y. Nagata, M. Obara, and K. Midorikawa, "Highly efficient, phase-matched high-harmonic generation by a self-guided laser beam," Phys. Rev. Lett. 82, 1422-1455 (1999).
[CrossRef]

Rev. Mod. Phys.

T. Brabec and F. Krausz, "Intense few-cycle laser fields: frontiers of nonlinear optics," Rev. Mod. Phys. 72, 545-591 (2000).
[CrossRef]

Sov. Phys. JETP

A. M. Perelemov, V. S. Popov, and M. V. Terent'ev, "Ionization of atoms in alternating electric field," Sov. Phys. JETP 23, 924-934 (2004).

Other

J.-C. Diels and W. Rudolf, Ultrashort Laser Phenomena (Academic, 1996).

Catalog (CVI Laser corporation, Albuquerque, New Mexico, 1997).

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Figures (9)

Fig. 1
Fig. 1

Schematic view of proposed spectral broadening using conjugate gradient method.

Fig. 2
Fig. 2

Comparison of the energy evolution of the pulse during propagation in free space and hollow fiber, obtained using conjugate pressure gradient method and constant gas pressure. The energy discontinuity is due to bad coupling between a beam and a hollow fiber. In the inset, the beam profile of the respective simulations are compared.

Fig. 3
Fig. 3

Evolution of the power spectrum inside a hollow fiber as a function of propagation distance.

Fig. 4
Fig. 4

Plot of the peak intensity during the propagation in a hollow fiber (top), and the corresponding effective plasma density (bottom).

Fig. 5
Fig. 5

Evolution of spatial and temporal profiles during propagation inside a hollow fiber.

Fig. 6
Fig. 6

Evolution of the power spectrum at z = 30 cm (dotted curve), z = 40 cm (dashed curve), and z = 50 cm (thin curve), obtained from simulations where the real part of nonlinearity is omitted (top). In the inset are the occupation probabilities of the three lower-order modes. Power spectrum at the hollow fiber exit ( z = 60 cm ) obtained from simulation with (thick curve) and without (thin curve) the instantaneous nonlinearity are compared (bottom).

Fig. 7
Fig. 7

Plot of energy transmittance, accumulated ionization and leakage loss, respectively, during the passage in hollow fiber (top). Fraction of the pulse energy occupying different eigenmodes (bottom).

Fig. 8
Fig. 8

Spatiospectral profile of the beam after propagating in vacuum to a distance 100 cm (top), and the corresponding spatiospectral profile (bottom).

Fig. 9
Fig. 9

Power spectrum at a circular area of radii 4 mm (top) and the corresponding beam shape (bottom).

Equations (6)

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2 E ( r , z , t ) 2 c 2 z t E ( r , z , t ) = 1 ϵ 0 c 2 2 t 2 P NL ( r , z , t ) ,
P NL ( r , z , t ) = p ( z ) { Δ χ ( r , z , t ) ω p 2 ω 0 2 } E ( r , z , t ) ,
p ( z ) = p max ( z L I ) 1 2 , 0 < z < L I ,
p ( z ) = p max [ ( L I + L D z ) L D ] 1 2 , L I < x < L I + L D ,
ρ t = N 0 ( 1 ρ ) Γ ( I ) ,
ρ ̃ ( z ) = 1 A T A ρ ( t ) d t d A ,

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