Abstract

The filamentational instability of spatially broadband femtosecond optical pulses in air is investigated by means of a kinetic wave equation for spatially incoherent photons. An explicit expression for the spatial amplification rate is derived and analyzed. It is found that the spatial spectral broadening of the pulse can lead to stabilization of the filamentation instability. Thus optical smoothing techniques could optimize current applications of ultrashort laser pulses, such as atmospheric remote sensing.

© 2006 Optical Society of America

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    [CrossRef]

2006 (2)

T. T. Xi, X. Lu, and J. Zhang, Phys. Rev. Lett. 96, 025003 (2006).
[CrossRef] [PubMed]

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

2005 (1)

A. Vinçotte and L. Bergé, Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef] [PubMed]

2004 (2)

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

2003 (1)

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, Phys. Lett. A 319, 317 (2003).
[CrossRef]

2002 (2)

A. Couairon and L. Bergé, Phys. Rev. Lett. 88, 135003 (2002).
[CrossRef] [PubMed]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

2001 (1)

J. T. Mendonça, Theory of Photon Acceleration (Institute of Physics, 2001).
[CrossRef]

2000 (1)

A. Couairon and L. Bergé, Phys. Plasmas 7, 193 (2000).
[CrossRef]

1999 (1)

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

1998 (2)

1996 (1)

1995 (1)

1994 (1)

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, Phys. Rev. E 50, R3314 (1994).
[CrossRef]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

1983 (1)

E. I. Ivleva, V. V. Korobkin, and V. N. Sazonov, Sov. J. Quantum Electron. 13, 754 (1983).
[CrossRef]

1981 (1)

V. V. Korobkin and V. N. Sazonov, Sov. Phys. JETP 54, 636 (1981).

1967 (1)

Yu. L. Klimontovich, The Statistical Theory of Non-Equilibrium Processes in a Plasma (Pergamon, 1967).

1949 (1)

J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).
[CrossRef]

1932 (1)

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Aleksic, N. B.

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, Phys. Lett. A 319, 317 (2003).
[CrossRef]

Anderson, D.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Batani, D.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, Phys. Rev. E 50, R3314 (1994).
[CrossRef]

Benuzzi, A.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, Phys. Rev. E 50, R3314 (1994).
[CrossRef]

Berezhiani, V. I.

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, Phys. Lett. A 319, 317 (2003).
[CrossRef]

Bergé, L.

A. Vinçotte and L. Bergé, Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

A. Couairon and L. Bergé, Phys. Rev. Lett. 88, 135003 (2002).
[CrossRef] [PubMed]

A. Couairon and L. Bergé, Phys. Plasmas 7, 193 (2000).
[CrossRef]

Bossi, S.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, Phys. Rev. E 50, R3314 (1994).
[CrossRef]

Boudenne, J. M.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, Phys. Rev. E 50, R3314 (1994).
[CrossRef]

Bourayou, R.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Braun, A.

Cheong, W. C.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Chiron, A.

Couairon, A.

A. Couairon and L. Bergé, Phys. Rev. Lett. 88, 135003 (2002).
[CrossRef] [PubMed]

A. Couairon and L. Bergé, Phys. Plasmas 7, 193 (2000).
[CrossRef]

Curley, P. F.

Du, D.

Faral, B.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, Phys. Rev. E 50, R3314 (1994).
[CrossRef]

Fedele, R.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

Franco, M. A.

Grillon, G.

Hall, B.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

Ivleva, E. I.

E. I. Ivleva, V. V. Korobkin, and V. N. Sazonov, Sov. J. Quantum Electron. 13, 754 (1983).
[CrossRef]

Kasparian, J.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Klimontovich, Yu. L.

Yu. L. Klimontovich, The Statistical Theory of Non-Equilibrium Processes in a Plasma (Pergamon, 1967).

Koenig, M.

M. Koenig, B. Faral, J. M. Boudenne, D. Batani, A. Benuzzi, and S. Bossi, Phys. Rev. E 50, R3314 (1994).
[CrossRef]

Kolesik, M.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

Korn, G.

Korobkin, V. V.

E. I. Ivleva, V. V. Korobkin, and V. N. Sazonov, Sov. J. Quantum Electron. 13, 754 (1983).
[CrossRef]

V. V. Korobkin and V. N. Sazonov, Sov. Phys. JETP 54, 636 (1981).

Lamouroux, B.

Lange, H. R.

Lederer, F.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Lin, Z. Y.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Lisak, M.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

Liu, X.

Low, D. K. Y.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Lu, X.

T. T. Xi, X. Lu, and J. Zhang, Phys. Rev. Lett. 96, 025003 (2006).
[CrossRef] [PubMed]

Méjean, G.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

Mendonça, J. T.

J. T. Mendonça, Theory of Photon Acceleration (Institute of Physics, 2001).
[CrossRef]

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Mlejnek, M.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, Opt. Lett. 23, 382 (1998).
[CrossRef]

Moh, K. J.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Moloney, J. V.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, Opt. Lett. 23, 382 (1998).
[CrossRef]

Mourou, G.

Moyal, J. E.

J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).
[CrossRef]

Mysyrowicz, A.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Nibbering, E. T. J.

Niu, H. B.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Peng, X.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Peschel, U.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Prade, B. S.

Ripoche, J.-F.

Rodriguez, M.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Salin, F.

Salmon, E.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

Sauerbrey, R.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Sazonov, V. N.

E. I. Ivleva, V. V. Korobkin, and V. N. Sazonov, Sov. J. Quantum Electron. 13, 754 (1983).
[CrossRef]

V. V. Korobkin and V. N. Sazonov, Sov. Phys. JETP 54, 636 (1981).

Semenov, V. E.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

Skarka, V.

V. Skarka, N. B. Aleksic, and V. I. Berezhiani, Phys. Lett. A 319, 317 (2003).
[CrossRef]

Skupin, S.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

Squier, J.

Tang, D. Y.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Vinçotte, A.

A. Vinçotte and L. Bergé, Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef] [PubMed]

Wigner, E. P.

E. P. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Wolf, J. P.

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

Wöste, L.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Wright, E. M.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

M. Mlejnek, E. M. Wright, and J. V. Moloney, Opt. Lett. 23, 382 (1998).
[CrossRef]

Xi, T. T.

T. T. Xi, X. Lu, and J. Zhang, Phys. Rev. Lett. 96, 025003 (2006).
[CrossRef] [PubMed]

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, Phys. Rev. Lett. 53, 1057 (1984).
[CrossRef]

Yu, J.

L. Bergé, S. Skupin, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. Lett. 92, 225002 (2004).
[CrossRef] [PubMed]

S. Skupin, L. Bergé, U. Peschel, F. Lederer, G. Méjean, J. Yu, J. Kasparian, E. Salmon, J. P. Wolf, M. Rodriguez, L. Wöste, R. Bourayou, and R. Sauerbrey, Phys. Rev. E 70, 046602 (2004).
[CrossRef]

Yuan, X.-C.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Zhang, J.

T. T. Xi, X. Lu, and J. Zhang, Phys. Rev. Lett. 96, 025003 (2006).
[CrossRef] [PubMed]

Zhang, L. S.

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

K. J. Moh, X.-C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, Appl. Phys. Lett. 88, 091103 (2006).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Coherent filamentation instability growth rate, given by Eq. (6) in the limit Δ 0 , plotted for different parameter values; all curves with I 0 = 0.5 , α = 1 , and K = 3 . Solid curve, standard filamentation instability growth rate for a nonlinear Schrödinger equation (i.e., ν = ϵ = γ = 0 ); thin dashed curve, ν = γ = 0 , ϵ = 0.5 ; thin dotted curve, ν = ϵ = 0 and γ = 0.5 ; thin dashed-dotted curve, ν = 0 and ϵ = γ = 0.5 ; thick dashed curve, ν = 0.1 and ϵ = γ = 0 ; thick dashed-dotted curve, ν = 0.1 and ϵ = γ = 1 2 .

Fig. 2
Fig. 2

Partially coherent filamentation instability growth rate, given by Eq. (6), plotted for different parameter values; all curves with I 0 = 0.75 , α = 1 , and K = 3 . Solid curve, standard filamentation instability growth rate for a nonlinear Schrödinger equation (i.e., Δ = ν = ϵ = γ = 0 ); thin solid curve, ν = ϵ = γ = 0 , Δ = 0.1 ; thin dashed curve, ϵ = γ = 0 , ν = 0.05 , Δ = 0.1 ; thin dotted curve, ν = 0.05 , γ = 0.1 , ϵ = 0 . The effects of the finite width of the background intensity distribution of the optical pulse, as well as the influence of the higher-order nonlinearity and losses, are clearly seen here.

Equations (6)

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i z ψ + 2 ψ + f ( ψ 2 ) ψ + i ν ψ 2 K 2 ψ = 0 ,
ρ ( z , r , p ) = 1 ( 2 π ) 2 d 2 ξ e i p ξ ψ * ( z , r + ξ 2 ) ψ ( z , r ξ 2 ) ,
z ρ + 2 p ρ + 2 f ( I ) sin ( 1 2 p ) ρ + 2 ν I K 1 cos ( 1 2 p ) ρ = 0 .
1 = d 2 p [ f ( I ¯ ) + i ν ( K 1 ) I ¯ K 2 ] ρ ¯ ( z , p k 2 ) [ f ( I ¯ ) i ν ( K 1 ) I ¯ K 2 ] ρ ¯ ( z , p + k 2 ) k + 2 k p 2 i ν I ¯ K 1 ,
k = 2 k p 0 + i ν ( K + 1 ) I ¯ K 1 + 2 i k Δ ± k 2 [ k 2 2 f ( I ¯ ) I ¯ ] ν 2 ( K 1 ) 2 I ¯ 2 K 2 ,
Γ = ν ( K + 1 ) I ¯ K 1 2 k Δ + k 2 [ 2 f ( I ¯ ) I ¯ k 2 ] + ν 2 ( K 1 ) 2 I ¯ 2 K 2 .

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