Abstract

Spatiotemporal self-focusing in nonlinear lossy media pushes ultrashort pulses towards a universal, non-solitary and non-conical light-bullet wave state defined by the medium solely, and characterized by maximum energy losses. Its stationary propagation relies on a balance between nonlinear losses and the refuelling effect of self-focusing. No balancing gain is required for stationarity. These purely lossy dissipative light-bullets can explain many aspects of the filamentary dynamics in nonlinear media with anomalous dispersion.

© 2010 Optical Society of America

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  1. F. Wise and P. Di Trapani, "The hunt for light bullets—Spatiotemporal solitons," Opt. Photon. News, 29-32 (2002).
  2. H. S. Eisenberg, R. Morandotti, and Y. Silberberg, "Kerr spatiotemporal self-Focusing in a planar glass waveguide," Phys. Rev. Lett. 82,043902 (2001).
    [CrossRef]
  3. R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional soliton dragging logic," Phys. Rev. A 52,3254-3278 (1995).
    [CrossRef] [PubMed]
  4. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15,1282-1284 (1990).
    [CrossRef] [PubMed]
  5. N. Akhmediev and J. M. Soto Crespo, "Generation of a train of three-dimensional optical solitons in a self focusing medium," Phys. Rev. A 47,1358-1364 (1993).
    [CrossRef] [PubMed]
  6. G. Fibich and B. Ilan, "Optical light bullets in a pure Kerr medium," Opt. Lett. 29,887-889 (2004).
    [CrossRef] [PubMed]
  7. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, "Self-channeling of high-peak-power femtosecond laser pulses in air," Opt. Lett. 20,73-75 (1995).
    [CrossRef] [PubMed]
  8. S. Henz and J. Herrmann, "Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach," Phys. Rev. E 53,4092-4097 (1996).
    [CrossRef]
  9. D. Moll and A. L. Gaeta, "Role of dispersion in multiple-collapse dynamics," Opt. Lett. 29, 995-997 (2004).
    [CrossRef] [PubMed]
  10. L. Bergé and S. Skupin, "Self-channeling of ultrashort laser pulses in materials with anomalous dispersion," Phys. Rev. E 71,065601 (2005).
    [CrossRef]
  11. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
    [CrossRef] [PubMed]
  12. M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92,253901 (2004).
    [CrossRef] [PubMed]
  13. A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004).
    [CrossRef] [PubMed]
  14. M. Kolesik and J. V. Moloney, "Self-healing femtosecond light filaments," Opt. Lett. 29,590-592 (2004).
    [CrossRef] [PubMed]
  15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93,153902 (2004).
    [CrossRef] [PubMed]
  16. M. A. Porras, A. Parola, and P. Di Trapani, "Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media," J. Opt. Soc. Am. B 22,1406-1413 (2005).
    [CrossRef]
  17. M. A. Porras and A. Parola, "Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses," Opt. Lett. 33,1738-1740 (2008).
    [CrossRef] [PubMed]
  18. D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, "Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption," Opt. Express 16,8213-8218 (2008).
    [CrossRef] [PubMed]
  19. N. Akhmediev and A. Ankiewicz (Eds.), Dissipative solitons, Lecture Notes in Physics 661 (Springer, 2005).
    [CrossRef]
  20. P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, "Light bullets and dynamic pattern formation in nonlinear dissipative systems," Opt. Express 13,9352-9360 (2005).
    [CrossRef] [PubMed]
  21. J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, "Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14,4013-4025 (2006).
    [CrossRef] [PubMed]
  22. W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77,023814 (2008).
    [CrossRef]
  23. Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, "Realization of a semiconductor-based cavity soliton laser," Phys. Rev. Lett. 100,013907 (2008).
    [CrossRef] [PubMed]
  24. This peak power is slightly higher than the power Pcr = 2λ20/[4πn(ω0)n2] for spatial self-focusing.

2008 (4)

2006 (1)

2005 (3)

2004 (6)

D. Moll and A. L. Gaeta, "Role of dispersion in multiple-collapse dynamics," Opt. Lett. 29, 995-997 (2004).
[CrossRef] [PubMed]

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92,253901 (2004).
[CrossRef] [PubMed]

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004).
[CrossRef] [PubMed]

M. Kolesik and J. V. Moloney, "Self-healing femtosecond light filaments," Opt. Lett. 29,590-592 (2004).
[CrossRef] [PubMed]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93,153902 (2004).
[CrossRef] [PubMed]

G. Fibich and B. Ilan, "Optical light bullets in a pure Kerr medium," Opt. Lett. 29,887-889 (2004).
[CrossRef] [PubMed]

2003 (1)

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

2002 (1)

F. Wise and P. Di Trapani, "The hunt for light bullets—Spatiotemporal solitons," Opt. Photon. News, 29-32 (2002).

2001 (1)

H. S. Eisenberg, R. Morandotti, and Y. Silberberg, "Kerr spatiotemporal self-Focusing in a planar glass waveguide," Phys. Rev. Lett. 82,043902 (2001).
[CrossRef]

1996 (1)

S. Henz and J. Herrmann, "Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach," Phys. Rev. E 53,4092-4097 (1996).
[CrossRef]

1995 (2)

1993 (1)

N. Akhmediev and J. M. Soto Crespo, "Generation of a train of three-dimensional optical solitons in a self focusing medium," Phys. Rev. A 47,1358-1364 (1993).
[CrossRef] [PubMed]

1990 (1)

Ackermann, T.

Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, "Realization of a semiconductor-based cavity soliton laser," Phys. Rev. Lett. 100,013907 (2008).
[CrossRef] [PubMed]

Akhmediev, N.

Averchi, A.

Bergé, L.

L. Bergé and S. Skupin, "Self-channeling of ultrashort laser pulses in materials with anomalous dispersion," Phys. Rev. E 71,065601 (2005).
[CrossRef]

Blair, S.

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional soliton dragging logic," Phys. Rev. A 52,3254-3278 (1995).
[CrossRef] [PubMed]

Bragheri, F.

Braun, A.

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77,023814 (2008).
[CrossRef]

Clerici, M.

Conti, C.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

Couairon, A.

Cristiani, I.

Di Trapani, P.

D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, "Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption," Opt. Express 16,8213-8218 (2008).
[CrossRef] [PubMed]

M. A. Porras, A. Parola, and P. Di Trapani, "Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media," J. Opt. Soc. Am. B 22,1406-1413 (2005).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93,153902 (2004).
[CrossRef] [PubMed]

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

F. Wise and P. Di Trapani, "The hunt for light bullets—Spatiotemporal solitons," Opt. Photon. News, 29-32 (2002).

Du, D.

Dubietis, A.

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004).
[CrossRef] [PubMed]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93,153902 (2004).
[CrossRef] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, R. Morandotti, and Y. Silberberg, "Kerr spatiotemporal self-Focusing in a planar glass waveguide," Phys. Rev. Lett. 82,043902 (2001).
[CrossRef]

Faccio, D.

Fibich, G.

Firth, W. J.

Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, "Realization of a semiconductor-based cavity soliton laser," Phys. Rev. Lett. 100,013907 (2008).
[CrossRef] [PubMed]

Gaeta, A. L.

Gaizauskas, E.

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004).
[CrossRef] [PubMed]

Grelu, P.

Henin, S.

Henz, S.

S. Henz and J. Herrmann, "Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach," Phys. Rev. E 53,4092-4097 (1996).
[CrossRef]

Herrmann, J.

S. Henz and J. Herrmann, "Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach," Phys. Rev. E 53,4092-4097 (1996).
[CrossRef]

Ilan, B.

Jäger, R.

Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, "Realization of a semiconductor-based cavity soliton laser," Phys. Rev. Lett. 100,013907 (2008).
[CrossRef] [PubMed]

Jedrkiewicz, O.

Kolesik, M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92,253901 (2004).
[CrossRef] [PubMed]

M. Kolesik and J. V. Moloney, "Self-healing femtosecond light filaments," Opt. Lett. 29,590-592 (2004).
[CrossRef] [PubMed]

Korn, G.

Liu, X.

McLeod, R.

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional soliton dragging logic," Phys. Rev. A 52,3254-3278 (1995).
[CrossRef] [PubMed]

Moll, D.

Moloney, J. V.

M. Kolesik and J. V. Moloney, "Self-healing femtosecond light filaments," Opt. Lett. 29,590-592 (2004).
[CrossRef] [PubMed]

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92,253901 (2004).
[CrossRef] [PubMed]

Morandotti, R.

H. S. Eisenberg, R. Morandotti, and Y. Silberberg, "Kerr spatiotemporal self-Focusing in a planar glass waveguide," Phys. Rev. Lett. 82,043902 (2001).
[CrossRef]

Mourou, G.

Papazoglou, D.

Parola, A.

Piskarskas, A.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

Porras, M. A.

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77,023814 (2008).
[CrossRef]

Silberberg, Y.

H. S. Eisenberg, R. Morandotti, and Y. Silberberg, "Kerr spatiotemporal self-Focusing in a planar glass waveguide," Phys. Rev. Lett. 82,043902 (2001).
[CrossRef]

Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15,1282-1284 (1990).
[CrossRef] [PubMed]

Skupin, S.

L. Bergé and S. Skupin, "Self-channeling of ultrashort laser pulses in materials with anomalous dispersion," Phys. Rev. E 71,065601 (2005).
[CrossRef]

Soto Crespo, J. M.

N. Akhmediev and J. M. Soto Crespo, "Generation of a train of three-dimensional optical solitons in a self focusing medium," Phys. Rev. A 47,1358-1364 (1993).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

Squier, J.

Tamosauskas, G.

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004).
[CrossRef] [PubMed]

Tanguy, Y.

Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, "Realization of a semiconductor-based cavity soliton laser," Phys. Rev. Lett. 100,013907 (2008).
[CrossRef] [PubMed]

Tartara, L.

Trillo, S.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

Trita, A.

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

Tzortzakis, S.

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

Wagner, K.

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional soliton dragging logic," Phys. Rev. A 52,3254-3278 (1995).
[CrossRef] [PubMed]

Wise, F.

F. Wise and P. Di Trapani, "The hunt for light bullets—Spatiotemporal solitons," Opt. Photon. News, 29-32 (2002).

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77,023814 (2008).
[CrossRef]

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92,253901 (2004).
[CrossRef] [PubMed]

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

Opt. Express (3)

Opt. Lett. (6)

Opt. Photon. News (1)

F. Wise and P. Di Trapani, "The hunt for light bullets—Spatiotemporal solitons," Opt. Photon. News, 29-32 (2002).

Phys. Rev. A (3)

N. Akhmediev and J. M. Soto Crespo, "Generation of a train of three-dimensional optical solitons in a self focusing medium," Phys. Rev. A 47,1358-1364 (1993).
[CrossRef] [PubMed]

W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77,023814 (2008).
[CrossRef]

R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional soliton dragging logic," Phys. Rev. A 52,3254-3278 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (2)

L. Bergé and S. Skupin, "Self-channeling of ultrashort laser pulses in materials with anomalous dispersion," Phys. Rev. E 71,065601 (2005).
[CrossRef]

S. Henz and J. Herrmann, "Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach," Phys. Rev. E 53,4092-4097 (1996).
[CrossRef]

Phys. Rev. Lett. (6)

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93,153902 (2004).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92,253901 (2004).
[CrossRef] [PubMed]

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004).
[CrossRef] [PubMed]

H. S. Eisenberg, R. Morandotti, and Y. Silberberg, "Kerr spatiotemporal self-Focusing in a planar glass waveguide," Phys. Rev. Lett. 82,043902 (2001).
[CrossRef]

Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, "Realization of a semiconductor-based cavity soliton laser," Phys. Rev. Lett. 100,013907 (2008).
[CrossRef] [PubMed]

Other (2)

This peak power is slightly higher than the power Pcr = 2λ20/[4πn(ω0)n2] for spatial self-focusing.

N. Akhmediev and A. Ankiewicz (Eds.), Dissipative solitons, Lecture Notes in Physics 661 (Springer, 2005).
[CrossRef]

Supplementary Material (1)

» Media 1: MOV (1189 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Peak intensity (solid curves) and FWHM width (dashed blue curves) along z for input Gaussian pulses of carrier frequency ω 0 = 1.21525 fs-1, Gaussian width σ = 0.00716 cm [duration σ(k 0k 0∣)1/2 = 29 fs], and increasing power above the critical power P cr = 13 MW, calculated from (1) with the parameters of fused silica (k 0 = 5.854 × 104 cm-1, k 0 = -279.4 cm-1fs2, n 2 = 2.2 × 10-16 cm2/W, β (K) = 5.11 × 10-116 cm17/W9, with K = 10[10]).

Fig. 2.
Fig. 2.

(a) Intensity profile a 2(r) (solid curves) and their asymptotic form (dotted blue curves), and (b) nonlinear losses profile (k 0k 0∣)1/2 N(r) (equal to the inward energy flux profile -(k 0k 0∣)1/2 F(r)) of DLBs in fused silica at λ0 = 1550 nm with increasing peak intensities I 0 = a 2(0). The intensity and loss profiles of the LB with maximum intensity I 0,m = 14.716 TW/cm2 [Eq. (4) with γK = 3.15294 for K = 10] are represented with dashed red curves. (c) Total losses (k 0k 0∣)1/2 NT of DLBs in fused silica at 1550 nm as a function of their peak intensity (solid curves). The total losses of conical LBs with δ = 2 and 7 cm-1 are shown with dashed curves. (d) Shape of the DLBs of maximum intensity and losses (dashed red curves), and their asymptotic forms b 2/ρ 2/(2K-3) (dotted blue curves). The radial profiles are obtained by solving (2) and (3) in the dimensionless form f + 2f /ρ + ϕ 2 f + = 0, 8πγ0 ρ f 2K ρ 2 = 4πρ 2 φ f 2, with f = a/I 0 1/2, ρ = (2k 0 2 n 2 I 0/n 0)1/2 r, and γ = n 0 β (K) IK-2 /(4n 2 k 0), and using the maximum values of γ leading to localized solutions, which are γK ≃ 0.88283,1.43838,1.83801,2.16571,2.45008,2.70474,2.93740, 3.15294 for K = 3,4,… 10, respectively.

Fig. 3.
Fig. 3.

(a) Peak intensity (solid curve), energy (dashed blue curve) and losses (dotted green curve) along z in the same conditions as in Fig. 1 except the Gaussian width σ = 0.011 cm [Gaussian duration σ(k 0k 0∣)1/2 =44.5 fs] and the peak power P = 50P cr. (b-d) Radial intensity profile of the pulse with increasing propagation distance z (solid curves), of the DLB of maximum losses (dashed red curve) and of the DLB fitting the pulse (open circles). A vertical off-set is introduced for clarity.

Fig. 4.
Fig. 4.

Left: For the same example as in Fig. 3, animation of the evolution along z of the radial intensity profile (solid curve) as it approaches and relaxes from the DLB of maximum losses (dashed red curve) (Media 1). Right: for the same example as in the left panel, visualization of the evolution as a trajectory of the pulse in the parameter space of DLBs and conical LBs in fused silica at 1550 nm. The inset shows a tiny region close to I 0,m .

Equations (6)

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z A = i 2 k 0 Δ A i k 0 2 t 2 A + i k 0 n 2 n 0 A 2 A β ( K ) 2 A 2 K 2 A ,
a + 2 a r + 2 k 0 δa φ 2 a + 2 k 0 2 n 2 n 0 a 3 = 0 ,
β ( K ) 4 π 0 r drr 2 a 2 K = 4 π r 2 k 0 a 2 φ ,
α ( 1 α ) b r 2 + 2 k 0 δb k 0 2 N T 2 16 π 2 b 3 r 4 ( α 1 ) + 2 k 0 2 n 2 b 3 n 0 r 2 α ~ 0 .
I 0 , m = ( 4 γ K n 2 k 0 n 0 β ( K ) ) 1 / ( K 2 ) ,
α ( 1 α ) b r 2 [ k 0 b 2 K 2 β ( K ) 3 2 ] 2 r 2 4 α ( K 1 ) + 2 k 0 2 n 2 b 3 n 0 r 2 α ~ 0 .

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