Abstract

The interaction dynamics of X-waves in an AlGaAs waveguide array is theoretically considered. The nonlinear discrete diffraction dynamics of a waveguide array mediates the generation of spatio-temporal X-waves from pulsed initial conditions. The interactions between co-propagating and counter-propagating X-waves are studied. For the co-propagating case, the initial phase relation between the X-waves determine the attractive or repulsive behavior of the X-wave interaction. For the counter-propagating case, the collisions between X-waves generate a nonlinear phase-shift. These dynamics show that X-waves interact in a manner similar to solitons.

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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. E. Recami, M. Zamboni-Rached, and H. E. Hernandez-Figueroa, Localized waves (Wiley, 2007).
  3. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003);
    [Crossref] [PubMed]
  4. 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]
  5. 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]
  6. D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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]
  7. C. Conti and 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]
  8. S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70, 235123 (2004).
    [Crossref]
  9. 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]
  10. D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
    [Crossref] [PubMed]
  11. J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007)
    [Crossref] [PubMed]
  12. K. Staliunas and M. Tlidi, “Hyperbolic Transverse Patterns in Nonlinear Optical Resonators,” Phys. Rev. Lett. 94, 133902 (2005);
    [Crossref] [PubMed]
  13. L. Mollenauer and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, (Springer, 2006);
  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]

2008 (1)

2007 (2)

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (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]

2006 (1)

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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]

2005 (1)

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

2004 (3)

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]

C. Conti and 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]

2003 (2)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (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]

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]

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]

Aitchison, J. S.

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]

Boyd, A. R.

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]

Bragheri, F.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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]

Christodoulides, D. N.

D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[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]

Conti, C.

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007)
[Crossref] [PubMed]

C. Conti and 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]

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, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003);
[Crossref] [PubMed]

Couairon, A.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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]

Cundiff, S. T.

Di Trapani, P.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (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]

Droulias, S.

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]

Dubietis, A.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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]

Eisenberg, H. S.

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]

Faccio, D.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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]

Frumker, E.

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]

Gordon, J.

L. Mollenauer and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, (Springer, 2006);

Greenleaf, J. F.

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]

Hernandez-Figueroa, H. E.

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

Hizanidis, K.

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]

Hudson, D.

Janner, D.

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

Jedrkiewicz, O.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (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]

Kolesik, M.

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]

Kutz, J. N.

Lahini, Y.

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]

Longhi, S.

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

Lu, J.

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]

Mollenauer, L.

L. Mollenauer and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, (Springer, 2006);

Moloney, J. V.

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]

Morandotti, R.

D. Hudson, K. Shish, T. R. Schibli, J. N. Kutz, D. N. Christodoulides, R. Morandotti, and S. T. Cundiff, “Nonlinear femtosecond pulse reshaping in waveguide arrays,” Opt. Lett. 33, 1440–1442 (2008).
[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]

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (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]

Porras, M.

D. Faccio, M. Porras, A. Dubietis, F. Bragheri, A. Couairon, and 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]

Recami, E.

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

Schibli, T. R.

Shish, K.

Silberberg, Y.

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]

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]

Staliunas, K.

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

Tlidi, M.

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

Trillo, S.

J. N. Kutz, C. Conti, and S. Trillo, “Mode-locked X-wave lasers,” Opt. Express 15, 16022–16028 (2007)
[Crossref] [PubMed]

C. Conti and 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]

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, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (2003);
[Crossref] [PubMed]

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]

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

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]

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

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

Zamboni-Rached, M.

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

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]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

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

Phys. Rev. Lett. (8)

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]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear Electromagnetic X-waves,” Phys. Rev. Lett. 90, 170406 (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]

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, and 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 and 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]

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]

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

Other (2)

L. Mollenauer and J. Gordon, Solitons in Optical Fibers: Fundamentals and Applications, (Springer, 2006);

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

Supplementary Material (4)

» Media 1: MOV (2284 KB)     
» Media 2: MOV (2388 KB)     
» Media 3: MOV (2223 KB)     
» Media 4: MOV (2484 KB)     

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Figures (8)

Fig. 1
Fig. 1

Pseudo-color plot of in-phase pulses interacting. The final result is a pair of closely-spaced but distinct X-waves. The white and green dotted lines indicate the center of the X-wave at Z = 0 in the 0th and 1st waveguide respectively. The white and green dashed lines represent the center of the X-wave at the present value of Z. Notice that the final separation is slightly larger than the initial separation of the pulses. ( Media 1)

Fig. 2
Fig. 2

Pseudo-color plot of out of-phase pulses interacting. In this case, the X-waves attract and form a pair of X-waves with a negligible delay in time. The white and green dotted lines denote the center of X-wave in the 0th and 1st waveguides at Z = 0, and the dashed lines represent the center of the X-wave at the present value of Z. ( Media 2)

Fig. 3
Fig. 3

Plot of the X-wave separation as a function of phase-difference for three different initial separations. The dashed lines show the initial separation and the solid lines of the same color show the final X-wave separation. Regardless of initial separation, X-waves with small phase-differences repel and X-waves with phase-differences near π attract.

Fig. 4
Fig. 4

Plot of the final X-wave separation for a pair of in-phase X-waves, shown in black, and π out-of-phase X-waves, shown in blue with an initial separation of ΔT = 1. For sufficiently high initial powers, the in-phase solutions repel and the out-of-phase solutions attract.

Fig. 5
Fig. 5

Pseudo-color plot of two unequally sized X-waves with Δθ = 0. The X-wave in waveguide 0 has amplitude 2.0 while the pulse in waveguide 1 has amplitude 1. The larger pulse dominates the dynamics and the attracts the smaller pulse. ( Media 3)

Fig. 6
Fig. 6

Pseudo-color plot of the collision of counter-propagating X-waves. The top series of plots shows the propagation of both the forward and backward X-waves at three snapshots in time. The bottom series of plots follows the propagation of the forward X-wave as it interacts with the backward X-wave. ( Media 4)

Fig. 7
Fig. 7

Plot of the phase shift in radians and time delay of the interacting X-waves. The magnitude of the phase shift is dependent upon the size of the X-wave, but the time-delay remains zero for all initial conditions.

Fig. 8
Fig. 8

Plot the phase shift of counter-propagating X-waves as a function of initial separation. The different colors correspond to initial conditions with η +, η = 1.50, 1.25, 1.00, 0.75, and 0.50 for blue, green, red, teal, and purple respectively.

Equations (5)

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i A n Z + D 2 2 A n T 2 + c ( A n 1 + A n + 1 ) + γ | A n | 2 A n = 0 ,
A 0 ( 0 , T ) = η 0 sech ( T ) and A 1 ( 0 , T ) = η 1 sech ( T + Δ T ) × exp ( i Δ θ ) .
i A n Z + i σ d A n d T + D 2 2 A n T 2 + γ ( | A n | 2 + 2 | B n | 2 ) A n + c ( A n 1 + A n + 1 ) = 0
i B n Z i σ d B n d T + D 2 2 A n T 2 + γ ( 2 | A n | 2 + | B n | 2 ) B n + c ( B n 1 + B n + 1 ) = 0
A 0 ( Z , 0 ) = η + sech ( Z + Δ Z ) and B 0 ( Z , 0 ) = η sech ( Z Δ Z ) .

Metrics