Abstract

Bragg soliton is a solution of the nonlinear coupled mode equations (NLCMEs) that are a non-integrable system. Nevertheless, we show that for a broad region of soliton parameters the interaction between two Bragg solitons in a uniform fiber Bragg grating can be accurately described by using a trial function based on the known solution for two-soliton interaction in the massive Thirring model (MTM). In this region the similar behavior of Bragg solitons and solitons of the Thirring model enables one to calculate explicitly the approximate asymptotic properties of the interaction between two Bragg solitons such as the shifts in locations and phases of the solitons as a result of the interaction. We have validated that the similar behavior of Bragg solitons and solitons of the MTM is obtained for a broad range of parameters of the interacting solitons that can also be realized experimentally. Since the NLCMEs are not an integrable system, there is a parameter regime in which the interaction between Bragg solitons does not resemble the elastic interaction between MTM solitons. We describe an interaction between two co-propagating Bragg solitons that causes the inversion of the propagation direction of one of the solitons.

© 2010 Optical Society of America

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  1. J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–134 (1988).
    [CrossRef] [PubMed]
  2. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  3. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [CrossRef] [PubMed]
  4. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
    [CrossRef]
  5. A. Rosenthal and M. Horowitz, “Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses,” Opt. Lett. 31, 1334–1336 (2006).
    [CrossRef] [PubMed]
  6. C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
    [CrossRef]
  7. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
    [CrossRef]
  8. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  9. N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18–23 (1999).
    [CrossRef]
  10. D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
    [CrossRef]
  11. W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
    [CrossRef]
  12. C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
    [CrossRef]
  13. I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
    [CrossRef]
  14. Y. P. Shapira and M. Horowitz, “Optical AND gate based on soliton interaction in a fiber Bragg grating,” Opt. Lett. 32, 1211–1213 (2007).
    [CrossRef] [PubMed]
  15. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
    [CrossRef]
  16. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [CrossRef] [PubMed]
  17. W. E. Thirring, “A soluble relativistic field theory,” Ann. Phys. (N.Y.) 3, 91–112 (1958).
    [CrossRef]
  18. E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
    [CrossRef]
  19. D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the solution of the massive Thirring model,” Lett. Nuovo Cimento Soc. Ital. Fis. 20, 325–331 (1977).
    [CrossRef]
  20. E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
    [CrossRef]
  21. Z. Toroker and M. Horowitz, “Optimized split-step method for modeling nonlinear pulse propagation in fiber Bragg gratings,” J. Opt. Soc. Am. B 25, 448–457 (2008).
    [CrossRef]

2008 (1)

2007 (1)

2006 (3)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
[CrossRef]

A. Rosenthal and M. Horowitz, “Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses,” Opt. Lett. 31, 1334–1336 (2006).
[CrossRef] [PubMed]

2003 (1)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[CrossRef]

2000 (1)

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

1999 (3)

1998 (1)

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

1997 (1)

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

1990 (1)

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

1989 (2)

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

1988 (1)

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

1977 (2)

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[CrossRef]

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the solution of the massive Thirring model,” Lett. Nuovo Cimento Soc. Ital. Fis. 20, 325–331 (1977).
[CrossRef]

1958 (1)

W. E. Thirring, “A soluble relativistic field theory,” Ann. Phys. (N.Y.) 3, 91–112 (1958).
[CrossRef]

Aceves, A. B.

N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18–23 (1999).
[CrossRef]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Agrawal, G. P.

Atai, J.

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[CrossRef]

de Sterke, C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18–23 (1999).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

Eggleton, B. J.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18–23 (1999).
[CrossRef]

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Horowitz, M.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

Kaup, D. J.

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the solution of the massive Thirring model,” Lett. Nuovo Cimento Soc. Ital. Fis. 20, 325–331 (1977).
[CrossRef]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Kuznetsov, E. A.

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[CrossRef]

Litchinitser, N. M.

Littler, I. C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[CrossRef]

Malomed, B. A.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[CrossRef]

Mikhailov, A. V.

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Mok, J. T.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

Neill, D. R.

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
[CrossRef]

Newell, A. C.

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the solution of the massive Thirring model,” Lett. Nuovo Cimento Soc. Ital. Fis. 20, 325–331 (1977).
[CrossRef]

Pelinovsky, D. E.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Rosenthal, A.

Shapira, Y. P.

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–134 (1988).
[CrossRef] [PubMed]

Slusher, R. E.

Thirring, W. E.

W. E. Thirring, “A soluble relativistic field theory,” Ann. Phys. (N.Y.) 3, 91–112 (1958).
[CrossRef]

Toroker, Z.

Tsoy, E. N.

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

Wabnitz, S.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Winful, H. G.

Zemlyanaya, E. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Ann. Phys. (N.Y.) (1)

W. E. Thirring, “A soluble relativistic field theory,” Ann. Phys. (N.Y.) 3, 91–112 (1958).
[CrossRef]

J. Opt. Soc. Am. B (4)

Lett. Nuovo Cimento Soc. Ital. Fis. (1)

D. J. Kaup and A. C. Newell, “On the Coleman correspondence and the solution of the massive Thirring model,” Lett. Nuovo Cimento Soc. Ital. Fis. 20, 325–331 (1977).
[CrossRef]

Nat. Phys. (1)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (2)

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
[CrossRef]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Phys. Rev. A (1)

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

Phys. Rev. E (3)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[CrossRef]

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E 59, 1267–1269 (1999).
[CrossRef]

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

Phys. Rev. Lett. (4)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

Theor. Math. Phys. (1)

E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the two-dimensional classical Thirring model,” Theor. Math. Phys. 30, 193–200 (1977).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Comparison between the intensities calculated by the trial function I ̂ s ( z , t ) (dashed curve) and by using a numerical simulation I s ( z , t ) (solid curve) before the interaction (a) at t = 0 , during the interaction (b) at t = 50 ( W s / V g )   s and (c) at t = 70 ( W s / V g ) , and after the interaction (d) at t = 105 ( W s / V g ) , where W s = 1.57   cm in the spatial width of the solitons and W s / V g = 76   ps . The soliton parameters are ρ 1 = ρ 2 = 0.2 , v 2 = 0.6 , v 1 = 0.45 , φ 1 , 0 = φ 2 , 0 = 0 .

Fig. 2
Fig. 2

Comparison between the amplitudes and phases of the forward U + and the backward U propagating waves that are calculated by the trial function (dashed curve) and by the numerical solution (solid curve) at the end of the interaction at t = 105 ( W s / V g ) , where W s = 1.57   cm and W s / V g = 76   ps . The soliton parameters are ρ 1 = ρ 2 = 0.2 , v 2 = 0.6 , v 1 = 0.45 , φ 1 , 0 = φ 2 , 0 = 0 .

Fig. 3
Fig. 3

Comparison between the intensity of the trial function I ̂ s ( z , t ) (dashed curve) and the intensity calculated by using a numerical simulation I s ( z , t ) (solid curve) before the interaction at (a) t = 0 , during the interaction at (b) t = 50 ( W s / V g ) and (c) t = 70 ( W s / V g ) , and after the interaction at (d) t = 105 ( W s / V g ) , where W s = 1.57   cm and W s / V g = 76   ps . The soliton parameters are ρ 1 = ρ 2 = 0.2 , v 2 = 0.6 , v 1 = 0.45 , φ 1 , 0 = 0 , and φ 2 , 0 = ( 10 / 9 ) π .

Fig. 4
Fig. 4

Comparison between the amplitudes and phases of the forward U + and the backward U propagating waves that are calculated by the trial function (dashed curve) and by the numerical solution (solid curve) at the end of the interaction at t = 105 ( W s / V g ) , where W s = 1.57   cm and W s / V g = 76   ps . The soliton parameters are ρ 1 = ρ 2 = 0.2 , v 2 = 0.6 , v 1 = 0.45 , φ 1 , 0 = 0 , and φ 2 , 0 = ( 10 / 9 ) π .

Fig. 5
Fig. 5

Comparison between the intensity of the trial function I ̂ s ( z , t ) (dashed curve) and the intensity calculated by using a numerical simulation I s ( z , t ) (solid curve) before the interaction at (a) t = 0 , during the interaction at (b) t = 30 ( W s / V g ) and (c) t = 55 ( W s / V g ) , and after the interaction at (d) t = 95 ( W s / V g ) . The soliton parameters are ρ 1 = ρ 2 = 0.8 , v 2 = 0.8 , v 1 = 0.6 , φ 1 , 0 = 0 , and φ 2 , 0 = 0 . W s = 3.065   mm is the spatial width of the shorter soliton and W s / V g = 14.8   ps .

Fig. 6
Fig. 6

Comparison between the peak powers of (a) soliton 1 and (b) soliton 2 as a function of normalized time after the collision by solving numerically the NLCMEs (solid curve) and by using the trial function based on MTM interaction (dashed curve). The soliton parameters are ρ 1 = ρ 2 = 0.8 , v 2 = 0.8 , v 1 = 0.6 , φ 1 , 0 = 0 , and φ 2 , 0 = 0 . W s = 3.065   mm is the spatial width of the shorter soliton and W s / V g = 14.8   ps .

Fig. 7
Fig. 7

(a) Three-dimensional and (b) two-dimensional plots of a collision between two solitary waves calculated by solving numerically the NLCMEs. After the interaction one of the solitons (soliton 2) changes its propagation direction. The soliton parameters are ρ 1 = ρ 2 = 1.2 , v 2 = 0.01 , v 1 = 0.0075 , φ 1 , 0 = φ 2 , 0 = 0 .

Fig. 8
Fig. 8

Collision between two Bragg solitons at (a) t = 0 , (b) t = 320 ( W s / V g ) , (c) t = 420 ( W s / V g ) , (d) t = 1745 ( W s / V g ) , where W s = 3.6   mm and W s / V g = 17.5   ps . The soliton parameters are ρ 1 = ρ 2 = 1.2 , v 2 = 0.01 , v 1 = 0.0075 , φ 1 , 0 = φ 2 , 0 = 0 . The arrows indicate the propagation direction of the solitons.

Fig. 9
Fig. 9

Comparison between the trial function and the numerical simulation for two Bragg soliton interaction calculated for several parameters of soliton 2 ( ρ 2 , v 2 ) . The parameters of the first soliton (soliton 1) were set to v 1 = 0.75 v 2 , ρ 1 = ρ 2 , and φ 1 , 0 = 0 . For each parameter set ( ρ 2 , v 2 ) , we have calculated the interaction for nine different phases of soliton 2, φ 2 , 0 , which were varied between 0 and 2 π with a step-size of 2 π / 9 . A parameter set ( ρ 2 , v 2 ) that gives at the end of the interaction an error smaller than 10% for both solitons for all the nine phases φ 2 , 0 that were checked is marked with a circle. Squares represent data points in which eight of the nine phases of φ 2 , 0 yielded a relative error less than 10%. Diamonds represent data points in which less than eight phases of φ 2 , 0 yielded relative error less than 10%.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

± i z u ± + i V g 1 t u ± + κ ( z ) u + ( Γ s | u ± | 2 + 2 Γ x | u | 2 ) u ± + δ ( z ) u ± = 0 ,
± i z χ ± + i V g 1 t χ ± + κ χ + 2 Γ | χ | 2 χ ± = 0.
χ ( z , t ) = K 1 ( z , t ) e i μ ( z , t ) | Γ | ,
χ + ( z , t ) = i [ ( z V g 1 t ) K 1 ( z , t ) ] e i μ ( z , t ) κ | Γ | ,
χ ± ( z , t ) = ± κ 2 Γ ( 1 ± v 1 v ) 1 / 4 sin ( ρ ) sech [ θ ( z , t ) i ρ / 2 ] e i σ ( z , t ) ,
θ ( z , t ) = γ κ   sin ( ρ ) ( z z 0 v V g t ) ,
σ ( z , t ) = γ κ   cos ( ρ ) [ v ( z z 0 ) V g t ] + φ 0 ,
u ± ( z , t ) = α χ ± ( z , t ) e i η ( z , t ) ,
α = ( 1 + 1 2 1 + v 2 1 v 2 ) 1 / 2 ,
η ( z , t ) = 2 α 2 γ 2 v   arctan [ tanh ( θ ) tan ( ρ 2 ) ] .
I p = max z ( | u + | 2 + | u | 2 ) = κ Γ α 2 γ 4 sin 2 ( ρ / 2 ) .
K 1 ( z , t ) = f 1 ( z , t ) + f 2 ( z , t ) ,
f 1 ( z , t ) κ 2 Δ 1   sin ( ρ 1 ) e i σ 1 ( z , t ) sech [ θ 1 ( z , t ) i ρ 1 2 ] ,
f 2 ( z , t ) κ 2 Δ 2   sin ( ρ 2 ) e i [ 2 ρ 1 + σ 2 ( z , t ) ] sech [ θ 2 ( z , t ) i ρ 2 2 ] .
f 1 ( z , t ) κ 2 Δ 1   sin ( ρ 1 ) e i σ 1 ( z , t ) Ω 2 | Ω 2 | sech [ θ ̂ 1 ( z , t ) i ρ 1 2 ] ,
f 2 ( z , t ) κ 2 Δ 2   sin ( ρ 2 ) e i [ 2 ρ 1 + σ 2 ( z , t ) ] | Ω 1 | Ω 1 sech [ θ ̂ 2 ( z , t ) i ρ 2 2 ] ,
θ ̂ 1 ( z , t ) = κ γ 1   sin ( ρ 1 ) { z [ z 1 , 0 ln | 1 / Ω 1 | κ γ 1   sin ( ρ 1 ) ] v 1 V g t } ,
θ ̂ 2 ( z , t ) = κ γ 2   sin ( ρ 2 ) { z [ z 2 , 0 + ln | 1 / Ω 2 | κ γ 2   sin ( ρ 2 ) ] v 2 V g t } .
Δ z 1 = ln | 1 / Ω 1 | κ γ 1   sin ( ρ 1 ) ,
Δ z 2 = ln | 1 / Ω 2 | κ γ 2   sin ( ρ 2 ) ,
Δ φ 1 = arg ( Ω 2 ) ,
Δ φ 2 = arg ( Ω 1 ) ,
u ̂ ( z , t ) = G 1 ( z , t ) e i μ ( z , t ) | Γ | ,
u ̂ + ( z , t ) = i G 2 ( z , t ) e i μ ( z , t ) κ | Γ | ,
G 1 ( z , t ) = α 1 f 1 ( z , t ) e i η 1 ( z , t ) + α 2 f 2 ( z , t ) e i [ η 2 ( z , t ) + 2 ρ 1 ] ,
G 2 ( z , t ) = α 1 [ ( z V g 1 t ) f 1 ( z , t ) ] e i η 1 ( z , t ) + α 2 [ ( z V g 1 t ) f 2 ( z , t ) ] e i [ η 2 ( z , t ) + 2 ρ 1 ] .
u ± ( z , t = 0 ) = u ± , s 1 ( z , t = 0 ) + u ± , s 2 ( z , t = 0 ) ,
z c m ( t 1 ) = z I ( z , t 1 ) d z I ( z , t 1 ) d z ,
ε i = max t 0 t t 1 { | I i ( t ) I ̂ i ( t ) I i ( t ) | , | v i ( t ) v ̂ i ( t ) v i ( t ) | } ,
χ ( z , t ) = K 1 ( z , t ) e i μ ( z , t ) | Γ | ,
χ + ( z , t ) = i [ ( z V g 1 t ) K 1 ( z , t ) ] e i μ ( z , t ) κ | Γ | ,
μ ( τ , τ + ) = 2 τ + | K 1 ( τ , τ ̃ + ) | 2 d τ ̃ + ,
τ ± = 1 2 ( z ± V g t ) .
K 1 ( z , t ) = f 1 ( z , t ) + f 2 ( z , t ) ,
f 1 ( z , t ) = [ 1 1 + 1 Ω 2 e i ρ 2 2 θ 2 cosh ( θ 1 i ρ 1 2 ) κ 2 Δ 1   sin ( ρ 1 ) e i σ 1 + e i ρ 2 2 θ 2 | Ω 1 | ( 1 + 1 Ω 2 e i ρ 2 2 θ 2 ) cosh ( θ 1 i ρ 1 2 ln | Ω 1 | ) D 1 ( z , t ) κ 2 Δ 1   sin ( ρ 1 ) e i σ 1 ] 1 ,
f 2 ( z , t ) = [ Ω 1 e i ρ 1 2 θ 1 1 + Ω 1 e i ρ 1 2 θ 1 cosh ( θ 2 i ρ 2 2 ) κ 2 Δ 2   sin ( ρ 2 ) e i σ 2 + Ω 1 | Ω 1 | ( 1 + Ω 1 e i ρ 1 2 θ 1 ) cosh ( θ 2 i ρ 2 2 ln | Ω 1 | ) D 2 ( z , t ) κ 2 Δ 2   sin ( ρ 2 ) e i σ 2 ] 1 .
D 1 ( z , t ) = 4 Δ 1 2 Δ 2 2   sin ( ρ 1 ) sin ( ρ 2 ) e i ( ρ 2 / 2 ) θ 2 | Δ 1 2 e i ρ 1 Δ 2 2 e i ρ 2 | 2 ( 1 + 1 Ω 2 e i ρ 2 2 θ 2 ) cos [ σ 1 σ 2 + ρ 1 ρ 2 2 i   log ( Ω ) ] ,
D 2 ( z , t ) = 4 Ω 1 Δ 1 2 Δ 2 2   sin ( ρ 1 ) sin ( ρ 2 ) e i ( ρ 1 / 2 ) θ 1 | Δ 1 2 e i ρ 1 Δ 2 2 e i ρ 2 | 2 ( 1 + Ω 1 e i ρ 1 2 θ 1 ) cos [ σ 1 σ 2 + ρ 1 ρ 2 2 i   log ( Ω ) ] ,
Ω = Δ 1 Δ 2 Δ 2 2 e i ρ 2 Δ 1 2 e i ρ 1 Δ 2 2 e i ρ 2 Δ 1 2 e i ρ 1 ,
θ i ( z , t ) = κ γ i   sin ( ρ i ) ( z z i , 0 v i V g t ) ,
σ i ( z , t ) = κ γ i   cos ( ρ i ) [ V g t v i ( z z i , 0 ) ] φ i , 0 ,
v i = 1 Δ i 4 1 + Δ i 4 ,
γ i = 1 1 v i 2 ,
Δ i 2 = ( 1 v i 1 + v i ) 1 / 2 ,
Ω i = ( Δ 3 i 2 e i ρ 3 i Δ i 2 e i ρ i Δ i 2 e i ρ i Δ 3 i 2 e i ρ 3 i ) 2 ,

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