Abstract

We theoretically demonstrate X-waves as global attractors that enable mode-locking of a laser cavity operating in the normal dispersion regime. This result is based upon a fully comprehensive physical model of the laser cavity, where the nonlinear discrete diffraction dynamics of a waveguide array mediates the spontaneous periodic generation of spatio-temporal X-waves.

© 2007 Optical Society of America

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References

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  1. J. Lu and J. F. Greenleaf, "Nondiffracting X waves-exact solutions to free-space scalar waveequation and their finite aperture realizations," IEEE Trans. Ultrason. Ferrelec. Freq. Contr. 39, 19-31 (1992).
    [CrossRef]
  2. P. Saari and K. Reivelt, "Evidence of X-Shaped Propagation-Invariant Localized Light Waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
    [CrossRef]
  3. E. Recami, M. Zamboni-Rached, and H.E. Hernandez-Figueroa, Localized waves (Wiley, 2007).
  4. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
    [CrossRef] [PubMed]
  5. C. Conti, "Generation and nonlinear dynamics of X-waves of the Schrodinger equation," Phys. Rev. E 70, 046613 (2004).
    [CrossRef]
  6. 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]
  7. M. Kolesik, E.M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92253901 (2004).
    [CrossRef] [PubMed]
  8. D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
    [CrossRef] [PubMed]
  9. C. Conti, S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice", Phys. Rev. Lett. 92, 120404 (2004).
    [CrossRef] [PubMed]
  10. S. Longhi and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
    [CrossRef]
  11. Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
    [CrossRef] [PubMed]
  12. S. Droulias, K. Hizanidis, J. Meier, and D. N. Christodoulides, "X-waves in nonlinear normally dispersive waveguide arrays," Opt. Exp. 13, 1827-1832 (2005).
    [CrossRef]
  13. Y. Kominis, N. Moshonas, P. Pagagiannis, K. Hizanidis, and D.N. Christodoulides, "Bessel X-waves in 2D and 3D bidispersive optical systems," Opt. Lett. 30, 2924 (2005).
    [CrossRef] [PubMed]
  14. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  15. D. N. Christodoulides, F. Lederer, Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices", Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  16. K. Staliunas and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
    [CrossRef] [PubMed]
  17. C.T. Zhou, M. Y. Yu, and X. T. He, "X-wave solutions of complex GL equation," Phys. Rev. E 73, 026209 (2006).
    [CrossRef]
  18. C. C. Cutler, "The Regenerative Pulse Generator," Proc. IRE 43, 140-148 (1955).
    [CrossRef]
  19. J. Levy and M. S. Sherwin, "Poincar sections of charge-density-wave dynamics: Mode locking," Phys. Rev. Lett. 67, 2846-2849 (1991).
    [CrossRef] [PubMed]
  20. H. Wang, R. de Paola, and W. I. Norwood, "Dynamics underlying the patterning of cardiac dysrhythmias," Phys. Rev. Lett. 70, 3671-3674 (1993).
    [CrossRef] [PubMed]
  21. M. I. Visscher and B. Rejaei, "Josephson Current through Charge DensityWaves," Phys. Rev. Lett. 79, 4461-4464 (1997).
    [CrossRef]
  22. B. Andreotti, "The Song of Dunes as a Wave-Particle Mode Locking," Phys. Rev. Lett. 93, 238001 (2004).
    [CrossRef] [PubMed]
  23. M. S. Paoletti and T. H. Solomon, "Front propagation and mode-locking in an advection-reaction-diffusion system," Phys. Rev. E 72, 046204 (2005).
    [CrossRef]
  24. S. M. Rezende, F. M. de Aguiar, R. L. Rodrguez-Surez, and A. Azevedo, "Mode Locking of Spin Waves Excited by Direct Currents in Microwave Nano-oscillators," Phys. Rev. Lett. 98, 087202 (2007).
    [CrossRef] [PubMed]
  25. S. T. Cundiff, J. M. Soto-Crespo and N. Akhmediev, "Experimental Evidence for Soliton Explosions," Phys. Rev. Lett. 88, 073903 (2002).
    [CrossRef] [PubMed]
  26. M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking," Phys. Rev. Lett. 97, 113902 (2006).
    [CrossRef] [PubMed]
  27. J. Javaloyes, J. Mulet, and S. Balle, "Passive Mode Locking of Lasers by Crossed-Polarization Gain Modulation," Phys. Rev. Lett. 97, 163902 (2006).
    [CrossRef] [PubMed]
  28. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
    [CrossRef]
  29. J. N. Kutz, "Mode-locked Soliton Lasers," SIAM Review 48, 629-678 (2006).
    [CrossRef]
  30. J. Proctor and J. N. Kutz, "Passive mode-locking by use of waveguide arrays," Opt. Lett. 13, 2013-2015 (2005).
    [CrossRef]
  31. J. Proctor and J. N. Kutz, "Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays," Opt. Express 13, 8933-8950 (2005).
    [CrossRef] [PubMed]
  32. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, 1990).
  33. M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity light bullets: 3D localized structures in nonlinear optical resonator," Phys. Rev. Lett. 93, 203901 (2004).
    [CrossRef] [PubMed]
  34. A. Chong, W. H. Renninger, and F. W. Wise, "All-normal-dispersion femtosecond fiber laser with pulse energy above 20nJ," Opt. Lett. 32, 2408-2410 (2007).
    [CrossRef] [PubMed]
  35. A. N. Khilo, E. G. Katranji, A. A. Ryzhevich, "Axicon-based Bessel resonator: analytical description and experiment," J. Opt. Soc. Am. A 18, 1986-1992 (2001).
    [CrossRef]
  36. J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," J. Mathematics and Computers in Simulation 74, 333-342 (2007).
    [CrossRef]

2007 (4)

S. M. Rezende, F. M. de Aguiar, R. L. Rodrguez-Surez, and A. Azevedo, "Mode Locking of Spin Waves Excited by Direct Currents in Microwave Nano-oscillators," Phys. Rev. Lett. 98, 087202 (2007).
[CrossRef] [PubMed]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," J. Mathematics and Computers in Simulation 74, 333-342 (2007).
[CrossRef]

A. Chong, W. H. Renninger, and F. W. Wise, "All-normal-dispersion femtosecond fiber laser with pulse energy above 20nJ," Opt. Lett. 32, 2408-2410 (2007).
[CrossRef] [PubMed]

2006 (5)

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking," Phys. Rev. Lett. 97, 113902 (2006).
[CrossRef] [PubMed]

J. Javaloyes, J. Mulet, and S. Balle, "Passive Mode Locking of Lasers by Crossed-Polarization Gain Modulation," Phys. Rev. Lett. 97, 163902 (2006).
[CrossRef] [PubMed]

J. N. Kutz, "Mode-locked Soliton Lasers," SIAM Review 48, 629-678 (2006).
[CrossRef]

C.T. Zhou, M. Y. Yu, and X. T. He, "X-wave solutions of complex GL equation," Phys. Rev. E 73, 026209 (2006).
[CrossRef]

2005 (6)

K. Staliunas and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, "Passive mode-locking by use of waveguide arrays," Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

M. S. Paoletti and T. H. Solomon, "Front propagation and mode-locking in an advection-reaction-diffusion system," Phys. Rev. E 72, 046204 (2005).
[CrossRef]

S. Droulias, K. Hizanidis, J. Meier, and D. N. Christodoulides, "X-waves in nonlinear normally dispersive waveguide arrays," Opt. Exp. 13, 1827-1832 (2005).
[CrossRef]

Y. Kominis, N. Moshonas, P. Pagagiannis, K. Hizanidis, and D.N. Christodoulides, "Bessel X-waves in 2D and 3D bidispersive optical systems," Opt. Lett. 30, 2924 (2005).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, "Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays," Opt. Express 13, 8933-8950 (2005).
[CrossRef] [PubMed]

2004 (6)

M. Kolesik, E.M. Wright, and J. V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92253901 (2004).
[CrossRef] [PubMed]

M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity light bullets: 3D localized structures in nonlinear optical resonator," Phys. Rev. Lett. 93, 203901 (2004).
[CrossRef] [PubMed]

C. Conti, S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice", Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

S. Longhi and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
[CrossRef]

C. Conti, "Generation and nonlinear dynamics of X-waves of the Schrodinger equation," Phys. Rev. E 70, 046613 (2004).
[CrossRef]

B. Andreotti, "The Song of Dunes as a Wave-Particle Mode Locking," Phys. Rev. Lett. 93, 238001 (2004).
[CrossRef] [PubMed]

2003 (3)

D. N. Christodoulides, F. Lederer, Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices", Nature 424, 817-823 (2003).
[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]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

2002 (1)

S. T. Cundiff, J. M. Soto-Crespo and N. Akhmediev, "Experimental Evidence for Soliton Explosions," Phys. Rev. Lett. 88, 073903 (2002).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

1998 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1997 (2)

M. I. Visscher and B. Rejaei, "Josephson Current through Charge DensityWaves," Phys. Rev. Lett. 79, 4461-4464 (1997).
[CrossRef]

P. Saari and K. Reivelt, "Evidence of X-Shaped Propagation-Invariant Localized Light Waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

1993 (1)

H. Wang, R. de Paola, and W. I. Norwood, "Dynamics underlying the patterning of cardiac dysrhythmias," Phys. Rev. Lett. 70, 3671-3674 (1993).
[CrossRef] [PubMed]

1992 (1)

J. Lu and J. F. Greenleaf, "Nondiffracting X waves-exact solutions to free-space scalar waveequation and their finite aperture realizations," IEEE Trans. Ultrason. Ferrelec. Freq. Contr. 39, 19-31 (1992).
[CrossRef]

1991 (1)

J. Levy and M. S. Sherwin, "Poincar sections of charge-density-wave dynamics: Mode locking," Phys. Rev. Lett. 67, 2846-2849 (1991).
[CrossRef] [PubMed]

1955 (1)

C. C. Cutler, "The Regenerative Pulse Generator," Proc. IRE 43, 140-148 (1955).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

IEEE Trans. Ultrason. Ferrelec. Freq. Contr. (1)

J. Lu and J. F. Greenleaf, "Nondiffracting X waves-exact solutions to free-space scalar waveequation and their finite aperture realizations," IEEE Trans. Ultrason. Ferrelec. Freq. Contr. 39, 19-31 (1992).
[CrossRef]

J. Mathematics and Computers in Simulation (1)

J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," J. Mathematics and Computers in Simulation 74, 333-342 (2007).
[CrossRef]

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

Nature (1)

D. N. Christodoulides, F. Lederer, Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices", Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Opt. Exp. (1)

S. Droulias, K. Hizanidis, J. Meier, and D. N. Christodoulides, "X-waves in nonlinear normally dispersive waveguide arrays," Opt. Exp. 13, 1827-1832 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. B (1)

S. Longhi and D. Janner, "X-shaped waves in photonic crystals," Phys. Rev. B 70, 235123 (2004).
[CrossRef]

Phys. Rev. E (3)

C. Conti, "Generation and nonlinear dynamics of X-waves of the Schrodinger equation," Phys. Rev. E 70, 046613 (2004).
[CrossRef]

M. S. Paoletti and T. H. Solomon, "Front propagation and mode-locking in an advection-reaction-diffusion system," Phys. Rev. E 72, 046204 (2005).
[CrossRef]

C.T. Zhou, M. Y. Yu, and X. T. He, "X-wave solutions of complex GL equation," Phys. Rev. E 73, 026209 (2006).
[CrossRef]

Phys. Rev. Lett. (18)

K. Staliunas and M. Tlidi, "Hyperbolic Transverse Patterns in Nonlinear Optical Resonators," Phys. Rev. Lett. 94, 133902 (2005).
[CrossRef] [PubMed]

J. Levy and M. S. Sherwin, "Poincar sections of charge-density-wave dynamics: Mode locking," Phys. Rev. Lett. 67, 2846-2849 (1991).
[CrossRef] [PubMed]

H. Wang, R. de Paola, and W. I. Norwood, "Dynamics underlying the patterning of cardiac dysrhythmias," Phys. Rev. Lett. 70, 3671-3674 (1993).
[CrossRef] [PubMed]

M. I. Visscher and B. Rejaei, "Josephson Current through Charge DensityWaves," Phys. Rev. Lett. 79, 4461-4464 (1997).
[CrossRef]

B. Andreotti, "The Song of Dunes as a Wave-Particle Mode Locking," Phys. Rev. Lett. 93, 238001 (2004).
[CrossRef] [PubMed]

S. M. Rezende, F. M. de Aguiar, R. L. Rodrguez-Surez, and A. Azevedo, "Mode Locking of Spin Waves Excited by Direct Currents in Microwave Nano-oscillators," Phys. Rev. Lett. 98, 087202 (2007).
[CrossRef] [PubMed]

S. T. Cundiff, J. M. Soto-Crespo and N. Akhmediev, "Experimental Evidence for Soliton Explosions," Phys. Rev. Lett. 88, 073903 (2002).
[CrossRef] [PubMed]

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking," Phys. Rev. Lett. 97, 113902 (2006).
[CrossRef] [PubMed]

J. Javaloyes, J. Mulet, and S. Balle, "Passive Mode Locking of Lasers by Crossed-Polarization Gain Modulation," Phys. Rev. Lett. 97, 163902 (2006).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete Spatial Optical Solitons in Waveguide Arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

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. 92253901 (2004).
[CrossRef] [PubMed]

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, P. Di Trapani, "Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

C. Conti, S. Trillo, "Nonspreading wave packets in three dimensions formed by an ultracold Bose gas in an optical lattice", Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef] [PubMed]

Y. Lahini, E. Frumker, Y. Silberberg, S. Droulias, K. Hizanidis, and D. N. Christodoulides, "Discrete X-Wave Formation in Nonlinear Waveguide Arrays," Phys. Rev. Lett. 98, 023901 (2007).
[CrossRef] [PubMed]

P. Saari and K. Reivelt, "Evidence of X-Shaped Propagation-Invariant Localized Light Waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, "Nonlinear Electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity light bullets: 3D localized structures in nonlinear optical resonator," Phys. Rev. Lett. 93, 203901 (2004).
[CrossRef] [PubMed]

Proc. IRE (1)

C. C. Cutler, "The Regenerative Pulse Generator," Proc. IRE 43, 140-148 (1955).
[CrossRef]

SIAM Review (1)

J. N. Kutz, "Mode-locked Soliton Lasers," SIAM Review 48, 629-678 (2006).
[CrossRef]

Other (2)

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, 1990).

E. Recami, M. Zamboni-Rached, and H.E. Hernandez-Figueroa, Localized waves (Wiley, 2007).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of a ring cavity configuration with passive (WGA for ML operation and a dispersive fiber section) and active (Erbium-doped fiber amplifier) elements. The input and output of the WGA are labeled A and B, respectively. The dominant physical effect is indicated for each component. The right graph demonstrates the ideal saturable absorption curve of the percentage of transmitted energy in the center wave-guide (E) versus intensity (I) generated by the WGA.

Fig. 2.
Fig. 2.

Temporal (top) and spectral (bottom) pulse shapes emerging at point B (see Fig. 1) in the central waveguide, showing formation of a ML pulse from white-noise initial data. Here Lf =1 m and La =3 mm, the average cavity GVD is normal (k″=15 ps2/km), and gain parameters are τ=0.1,e0 =1, and g0 =3.5.

Fig. 3.
Fig. 3.

Input (A) and output (B) temporal power distribution in the WGA. At the input, energy is only launched in the center waveguide (A0 ), while at the output the energy has spontaneously formed into the X-wave configuration involving about eleven guides. Only energy in the A0 mode is preserved upon re-entry into the fiber section of the cavity.

Fig. 4.
Fig. 4.

Time-domain profiles and its two-dimensional Fourier transform at the output (B) in the WGA after steady-state ML has been achieved. The X-wave structure is clearly seen in the topographical plot (top) of the output time-domain profiles of Fig. 3. Further, the expected wavenumber versus frequency dependence in the X-wave is shown in the Fourier domain (bottom).

Fig. 5.
Fig. 5.

Evolution to the steady-state output (B) in the neighboring waveguides A 1, A 2, and A 3. The bottom right graph is a bar graph of the steady-state distribution of energy (∫∞ -∞ |Aj |2 dT) in the waveguides. The symmetry about the center waveguide results from the initial condition being applied only in this waveguide. Note the significant redistribution of energy in the waveguides.

Equations (3)

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i Q Z 1 2 2 Q T 2 + Q 2 Q + i α Q ig ( Z ) ( 1 + τ 2 T 2 ) Q = 0 ,
g ( Z ) = 2 g 0 1 + Q 2 e 0 .
i d A n d Z ̂ + C ( A n 1 + A n + 1 ) + β A n 2 A n = 0 ,

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