Abstract

Breathing solitons, as localized wave packets with a periodic evolution in amplitude and duration, are able to model extreme wave events in complex nonlinear dispersive systems. We have numerically studied the formation and manipulation of graded-index breathing solitons embedded in nonlinear multimode fibers based on a single nonlinear Schrödinger equation that includes the spatial self-imaging effect through a periodically varying nonlinear parameter. Through changing specific parameters of the input optical field, we can manipulate the period and depth of graded-index breathing soliton dynamics under different relative strengths between the dispersion length and the self-imaging period of the multimode fiber. Our study can explicitly derive a robust mechanism to control the behavior of the breathing localized structure directly and contribute to a better understanding of the much more complex nonlinear graded-index soliton dynamics in multimode fibers.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (7)

W. Fu, L. G. Wright, P. Sidorenko, S. Backus, and F. W. Wise, “Several new directions for ultrafast fiber lasers [Invited],” Opt. Express 26(8), 9432–9463 (2018).
[Crossref] [PubMed]

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

J. Lægsgaard, “Spatial beam cleanup by pure Kerr processes in multimode fibers,” Opt. Lett. 43(11), 2700–2703 (2018).
[Crossref] [PubMed]

C. Mas Arabí, A. Kudlinski, A. Mussot, and M. Conforti, “Geometric parametric instability in periodically modulated graded-index multimode fibers,” Phys. Rev. A (Coll. Park) 97(2), 023803 (2018).
[Crossref]

A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, F. Copie, P. Szriftgiser, and S. Trillo, “Fibre multi-wave mixing combs reveal the broken symmetry of Fermi–Pasta–Ulam recurrence,” Nat. Photonics 12(5), 303–308 (2018).
[Crossref]

A. S. Ahsan and G. P. Agrawal, “Graded-index solitons in multimode fibers,” Opt. Lett. 43(14), 3345–3348 (2018).
[Crossref] [PubMed]

X. Zhong, L. Chen, K. Cheng, N. Yao, and J. Sheng, “Generation of single or double parallel breathing soliton pairs, bound breathing solitons, moving breathing solitons, and diverse composite breathing solitons in optical fibers,” Opt. Express 26(12), 15683–15692 (2018).
[Crossref] [PubMed]

2017 (7)

M. Conforti, C. Mas Arabi, A. Mussot, and A. Kudlinski, “Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers,” Opt. Lett. 42(19), 4004–4007 (2017).
[Crossref] [PubMed]

M. Yu, J. K. Jang, Y. Okawachi, A. G. Griffith, K. Luke, S. A. Miller, X. Ji, M. Lipson, and A. L. Gaeta, “Breather soliton dynamics in microresonators,” Nat. Commun. 8, 14569 (2017).
[Crossref] [PubMed]

E. Lucas, M. Karpov, H. Guo, M. L. Gorodetsky, and T. J. Kippenberg, “Breathing dissipative solitons in optical microresonators,” Nat. Commun. 8(1), 736 (2017).
[Crossref] [PubMed]

R. Dupiol, A. Bendahmane, K. Krupa, J. Fatome, A. Tonello, M. Fabert, V. Couderc, S. Wabnitz, and G. Millot, “Intermodal modulational instability in graded-index multimode optical fibers,” Opt. Lett. 42(17), 3419–3422 (2017).
[Crossref] [PubMed]

H. Guo, E. Lucas, M. H. Pfeiffer, M. Karpov, M. Anderson, J. Liu, M. Geiselmann, J. D. Jost, and T. J. Kippenberg, “Intermode breather solitons in optical microresonators,” Phys. Rev. X 7(4), 041055 (2017).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

2016 (4)

L. G. Wright, Z. Liu, D. A. Nolan, M. J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guénard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, and S. Wabnitz, “Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber,” Opt. Lett. 41(24), 5785–5788 (2016).
[Crossref] [PubMed]

M. Conforti, A. Mussot, A. Kudlinski, S. Rota Nodari, G. Dujardin, S. De Biévre, A. Armaroli, and S. Trillo, “Heteroclinic structure of parametric resonance in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 117(1), 013901 (2016).
[Crossref] [PubMed]

2015 (5)

M. Shen, J. Gao, and L. Ge, “Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media,” Sci. Rep. 5(1), 9814 (2015).
[Crossref] [PubMed]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115(22), 223902 (2015).
[Crossref] [PubMed]

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

C. Bao, L. Zhang, L. C. Kimerling, J. Michel, and C. Yang, “Soliton breathing induced by stimulated Raman scattering and self-steepening in octave-spanning Kerr frequency comb generation,” Opt. Express 23(14), 18665–18670 (2015).
[Crossref] [PubMed]

2014 (1)

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
[Crossref]

2013 (5)

2012 (1)

2011 (3)

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[Crossref] [PubMed]

Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express 19(18), 17298–17307 (2011).
[Crossref] [PubMed]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106(20), 204502 (2011).
[Crossref] [PubMed]

2008 (2)

2007 (1)

2005 (1)

2004 (1)

S. Longhi and D. Janner, “Self-focusing and nonlinear periodic beams in parabolic index optical fibres,” J. Opt. B: Quantum Semiclass 6(5), S303–S308 (2004).
[Crossref]

2001 (1)

A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute bose-einstein condensates,” Phys. Rev. Lett. 86(11), 2353–2356 (2001).
[Crossref] [PubMed]

2000 (1)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref] [PubMed]

1998 (1)

S. Flach and C. R. Willis, “Discrete breathers,” Phys. Rep. 295(5), 181–264 (1998).
[Crossref]

1997 (1)

1996 (1)

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68(2), 423–444 (1996).
[Crossref]

1993 (1)

1992 (1)

1991 (1)

A. Hasegawa and Y. Kodama, “Guiding-center soliton,” Phys. Rev. Lett. 66(2), 161–164 (1991).
[Crossref] [PubMed]

Abdullaev, F. K.

Agrawal, G. P.

Ahsan, A. S.

Akhmediev, N.

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106(20), 204502 (2011).
[Crossref] [PubMed]

Anderson, D.

Anderson, M.

H. Guo, E. Lucas, M. H. Pfeiffer, M. Karpov, M. Anderson, J. Liu, M. Geiselmann, J. D. Jost, and T. J. Kippenberg, “Intermode breather solitons in optical microresonators,” Phys. Rev. X 7(4), 041055 (2017).
[Crossref]

Antonelli, C.

Armaroli, A.

M. Conforti, A. Mussot, A. Kudlinski, S. Rota Nodari, G. Dujardin, S. De Biévre, A. Armaroli, and S. Trillo, “Heteroclinic structure of parametric resonance in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 117(1), 013901 (2016).
[Crossref] [PubMed]

Backus, S.

Bao, C.

Barthélémy, A.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref] [PubMed]

Belic, M.

Bendahmane, A.

Bischoff, S.

Broky, J.

Caravaca-Aguirre, A. M.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
[Crossref]

Chabchoub, A.

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106(20), 204502 (2011).
[Crossref] [PubMed]

Chbat, M. W.

Chen, L.

Chen, Z.

Cheng, K.

Christodoulides, D. N.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref] [PubMed]

L. G. Wright, Z. Liu, D. A. Nolan, M. J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115(22), 223902 (2015).
[Crossref] [PubMed]

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Conforti, M.

C. Mas Arabí, A. Kudlinski, A. Mussot, and M. Conforti, “Geometric parametric instability in periodically modulated graded-index multimode fibers,” Phys. Rev. A (Coll. Park) 97(2), 023803 (2018).
[Crossref]

A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, F. Copie, P. Szriftgiser, and S. Trillo, “Fibre multi-wave mixing combs reveal the broken symmetry of Fermi–Pasta–Ulam recurrence,” Nat. Photonics 12(5), 303–308 (2018).
[Crossref]

M. Conforti, C. Mas Arabi, A. Mussot, and A. Kudlinski, “Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers,” Opt. Lett. 42(19), 4004–4007 (2017).
[Crossref] [PubMed]

M. Conforti, A. Mussot, A. Kudlinski, S. Rota Nodari, G. Dujardin, S. De Biévre, A. Armaroli, and S. Trillo, “Heteroclinic structure of parametric resonance in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 117(1), 013901 (2016).
[Crossref] [PubMed]

Copie, F.

A. Mussot, C. Naveau, M. Conforti, A. Kudlinski, F. Copie, P. Szriftgiser, and S. Trillo, “Fibre multi-wave mixing combs reveal the broken symmetry of Fermi–Pasta–Ulam recurrence,” Nat. Photonics 12(5), 303–308 (2018).
[Crossref]

Couderc, V.

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

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L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115(22), 223902 (2015).
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Wagner, K.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
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Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
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S. Flach and C. R. Willis, “Discrete breathers,” Phys. Rep. 295(5), 181–264 (1998).
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W. Fu, L. G. Wright, P. Sidorenko, S. Backus, and F. W. Wise, “Several new directions for ultrafast fiber lasers [Invited],” Opt. Express 26(8), 9432–9463 (2018).
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L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
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L. G. Wright, Z. Liu, D. A. Nolan, M. J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
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L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
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L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115(22), 223902 (2015).
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L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
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W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
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H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68(2), 423–444 (1996).
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W. Fu, L. G. Wright, P. Sidorenko, S. Backus, and F. W. Wise, “Several new directions for ultrafast fiber lasers [Invited],” Opt. Express 26(8), 9432–9463 (2018).
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L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
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L. G. Wright, Z. Liu, D. A. Nolan, M. J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115(22), 223902 (2015).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
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L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
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Yang, C.

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W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
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J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014).
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O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12(6), 368–374 (2018).
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L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
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K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
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L. G. Wright, Z. Liu, D. A. Nolan, M. J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
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J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
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W. Fu, L. G. Wright, P. Sidorenko, S. Backus, and F. W. Wise, “Several new directions for ultrafast fiber lasers [Invited],” Opt. Express 26(8), 9432–9463 (2018).
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L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
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C. Bao, L. Zhang, L. C. Kimerling, J. Michel, and C. Yang, “Soliton breathing induced by stimulated Raman scattering and self-steepening in octave-spanning Kerr frequency comb generation,” Opt. Express 23(14), 18665–18670 (2015).
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X. Zhong, L. Chen, K. Cheng, N. Yao, and J. Sheng, “Generation of single or double parallel breathing soliton pairs, bound breathing solitons, moving breathing solitons, and diverse composite breathing solitons in optical fibers,” Opt. Express 26(12), 15683–15692 (2018).
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A. S. Ahsan and G. P. Agrawal, “Graded-index solitons in multimode fibers,” Opt. Lett. 43(14), 3345–3348 (2018).
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K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guénard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, and S. Wabnitz, “Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber,” Opt. Lett. 41(24), 5785–5788 (2016).
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L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115(22), 223902 (2015).
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E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
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Figures (5)

Fig. 1
Fig. 1 Simulations of multimode breathing soliton dynamics. Temporal (top) and spectral (bottom) evolutions of the fundamental multimode soliton shedding from Airy pulses. Left column corresponds to the case of q = 100 while right column to q = 1. (b) and (d) show the peak power evolutions of the breathing solitons. In both cases, intensity is color-coded on a 30 dB scale. The periodic compression (CP) and stretching (SP) over one breathing period are indicated by a black dashed line and a red one, respectively.
Fig. 2
Fig. 2 Temporal and spectral profiles of the breathing soliton. The shortest and the longest temporal profiles (a) and the corresponding spectral profiles (b) for a multimode breathing soliton over one breathing cycle for the case shown in Fig. 1(a). The blue (brick-red) curve in the figure corresponds to the temporal and spectral profiles at the maximal compression (stretching) instant when the breathing soliton is excited.
Fig. 3
Fig. 3 Peak powers of the breathing soliton as a function of the propagation distance in terms of dispersion length for different launched peak powers R (a) and selected truncation coefficients a (b).
Fig. 4
Fig. 4 Temporal (top) and spectral (bottom) evolutions over 50LD for the interaction between both in-phase Airy pulses inside a GRIN MMF with q = 100 (a, c) and q = 1 (b, d). The intensity is coded on a decibel scale using dimensionless variables.
Fig. 5
Fig. 5 Breathing solitons with a high breathing frequency and depth formed by the interaction between in-phase Airy pulses with varying time intervals in GRIN MMF. Temporal (a) and spectral (b) evolution of interaction between both Airy pulses with T1 = −1. (c) Peak power of pulses versus propagation distance for different time intervals.

Equations (10)

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E z = i 2 k 0 ( 2 E x 2 + 2 E y 2 ) i β 2 2 2 E T 2 i k 0 Δ r 2 ( x 2 + y 2 ) E + i n 2 ω 0 c | E | 2 E
| F ( x , y , z ) | = w 0 w ( z ) exp [ 1 2 ( x 2 + y 2 ) w 2 ( z ) ]
w 2 ( z ) = w 0 2 [ cos 2 ( π z / Z p ) + 1 p ( k 0 2 w 0 4 ) / Z p sin 2 ( π z / Z p ) ]
A z = i β 2 2 2 A T 2 + i γ ( z ) | A | 2 A
ξ = z / L D , τ = T / T 0 , U ( ξ , τ ) = A ( z , T ) / P 0
U ξ = i 1 2 2 U τ 2 + i N 2 g ( ξ ) | U | 2 U
g ( ξ ) = 1 / [ cos 2 ( π q ξ ) + C sin 2 ( π q ξ ) ] , q = L D / Z p , C = 1 p ( k 0 2 w 0 4 ) / Z p
U ( ξ = 0 , τ ) = R K ( a ) A i ( τ ) exp ( a τ )
E = + | U ( ξ = 0 , τ ) | 2 d τ = R K ( a ) 1 8 π a exp ( 2 a 3 3 ) ,
U ( ξ = 0 , τ ) = R K ( a ) { A i ( τ T 1 ) exp [ a ( τ T 1 ) ] + A i [ ( τ + T 1 ) ] exp [ a ( τ + T 1 ) ] } ,

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