Abstract

We introduce the generalized Lugiato-Lefever equation describing nonlinear effects in the bottle microresonators. We demonstrate that the nonlinear modes of these resonators can form multiple coexisting and overlapping nonlinear resonances and that their instabilities lead to the generation of the low repetition rate frequency combs.

© 2017 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2016 (6)

2015 (3)

2014 (2)

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

2013 (1)

M. Sumetsky, “Delay of light in an optical bottle resonator with nanoscale radius variation: dispersionless, broadband, and low loss,” Phys. Rev. Lett. 111, 163901 (2013).
[Crossref] [PubMed]

2011 (4)

2010 (1)

2009 (1)

M. Pollinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[Crossref] [PubMed]

2006 (1)

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2212 (1987).
[Crossref] [PubMed]

Asano, M.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, “Stimulated Brillouin scattering and Brillouin-coupled four-wave-mixing in a silica microbottle resonator,” Opt. Express 24, 2082–2092 (2016).
[Crossref]

Ballato, J.

Barucci, A.

Brasch, V.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

Chormaic, S. N.

Coen, S.

Conti, G. N.

Dibbs, A. N.

Dvoyrin, V.

Erkintalo, M.

Farnesi, D.

Fini, J. M.

Franz, Y.

Freude, W.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Gibson, U. J.

Gorodetsky, M. L.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

Hansson, T.

Hartinger, K.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Healy, N.

Herr, T.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

Hillerkuss, D.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Holzwarth, R.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Ikuta, R.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, “Stimulated Brillouin scattering and Brillouin-coupled four-wave-mixing in a silica microbottle resonator,” Opt. Express 24, 2082–2092 (2016).
[Crossref]

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nature Photon. 5, 293–296 (2011).
[Crossref]

Imoto, N.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, “Stimulated Brillouin scattering and Brillouin-coupled four-wave-mixing in a silica microbottle resonator,” Opt. Express 24, 2082–2092 (2016).
[Crossref]

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

Kippenberg, T. J.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

Koos, C.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Laegsgaard, J.

Lauermann, M.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Lefever, R.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2212 (1987).
[Crossref] [PubMed]

Leuthold, J.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Li, J. S.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Liang, W.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nature Photon. 5, 293–296 (2011).
[Crossref]

Liu, L.

Liu, S.

Lu, Q.

Lugiato, L. A.

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2212 (1987).
[Crossref] [PubMed]

Maleki, L.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nature Photon. 5, 293–296 (2011).
[Crossref]

Matsko, A. B.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nature Photon. 5, 293–296 (2011).
[Crossref]

Murdoch, S. G.

O’Shea, D.

M. Pollinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[Crossref] [PubMed]

Ooka, Y.

Ozdemir, S. K.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, “Stimulated Brillouin scattering and Brillouin-coupled four-wave-mixing in a silica microbottle resonator,” Opt. Express 24, 2082–2092 (2016).
[Crossref]

Peacock, A. C.

Pfeifle, J.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Pollinger, M.

M. Pollinger and A. Rauschenbeutel, “All-optical signal processing at ultra-low powers in bottle microresonators using the Kerr effect,” Opt. Express 18, 17764–17775 (2010).
[Crossref] [PubMed]

M. Pollinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[Crossref] [PubMed]

Rauschenbeutel, A.

M. Pollinger and A. Rauschenbeutel, “All-optical signal processing at ultra-low powers in bottle microresonators using the Kerr effect,” Opt. Express 18, 17764–17775 (2010).
[Crossref] [PubMed]

M. Pollinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[Crossref] [PubMed]

Righini, G. C.

Savchenkov, A. A.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nature Photon. 5, 293–296 (2011).
[Crossref]

Schindler, P.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Schmogrow, R.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Seidel, D.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nature Photon. 5, 293–296 (2011).
[Crossref]

Skryabin, D. V.

Soria, S.

Suh, M. G.

M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref] [PubMed]

Suhailin, F. H.

Sumetsky, M.

Takeuchi, Y.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, “Stimulated Brillouin scattering and Brillouin-coupled four-wave-mixing in a silica microbottle resonator,” Opt. Express 24, 2082–2092 (2016).
[Crossref]

Thompson, R. M.

Vahala, K. J.

M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref] [PubMed]

Wabnitz, S.

Wang, C. Y.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

Ward, J. M.

Warken, F.

M. Pollinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultrahigh-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[Crossref] [PubMed]

Webb, K. E.

Wegner, D.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Weimann, C.

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Wu, X.

Xu, L.

Yamamoto, T.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, “Stimulated Brillouin scattering and Brillouin-coupled four-wave-mixing in a silica microbottle resonator,” Opt. Express 24, 2082–2092 (2016).
[Crossref]

Yang, K. Y.

M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref] [PubMed]

Yang, L.

M. Asano, Y. Takeuchi, S. K. Ozdemir, R. Ikuta, L. Yang, N. Imoto, and T. Yamamoto, “Stimulated Brillouin scattering and Brillouin-coupled four-wave-mixing in a silica microbottle resonator,” Opt. Express 24, 2082–2092 (2016).
[Crossref]

Yang, Q. F.

M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref] [PubMed]

Yang, Y.

Yi, X.

M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref] [PubMed]

Yimin, Y. Yu

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

Yulin, A. V.

J. Opt. Soc. Am. B (3)

Nature Photon. (3)

J. Pfeifle, V. Brasch, M. Lauermann, Y. Yu Yimin, D. Wegner, T. Herr, K. Hartinger, P. Schindler, J. S. Li, D. Hillerkuss, R. Schmogrow, C. Weimann, R. Holzwarth, W. Freude, J. Leuthold, T. J. Kippenberg, and C. Koos, “Coherent terabit communications with microresonator Kerr frequency combs,” Nature Photon. 8, 375–380 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nature Photon. 8, 145–152 (2014).
[Crossref]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nature Photon. 5, 293–296 (2011).
[Crossref]

Opt. Express (4)

Opt. Lett. (6)

Phys. Rev. Lett. (3)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2212 (1987).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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Science (1)

M. G. Suh, Q. F. Yang, K. Y. Yang, X. Yi, and K. J. Vahala, “Microresonator soliton dual-comb spectroscopy,” Science 354, 600–603 (2016).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (a) Density plot showing the spatial profiles vs the eigen-frequency ϵ of the 50 modes in the linear bottle resonator. (b) Spatial profiles, | ψ ˜ ( z ) |, of the upper branches of the nonlinear modes numerically computed using Eq. (2) with t = 0: zp = 0, P0 = 1, κ = 10−3; δ = 0 (n = 0, 2), δ = −10 (n = 10, 12). (c) Geometry of a bottle resonator.
Fig. 2
Fig. 2 (a–c) Structure of the multiple nonlinear resonances for the different pump positions and the resonator losses. Plots show I = | ψ ˜ ( z ) | 2 d z vs δ. (a) zp = 0 (maximum of the ground state mode), κ = 10−3, (b) zp = 0.79 (maximum of the n = 1 mode), κ = 10−3, (c) κ = 10−1, zp = 0. The bold points show the onsets of the instabilities of the upper branches of the nonlinear resonances, which are stable below these points. The encircled numbers correspond to the modal index n. (d) The instability growth rates Re λ vs δ of the upper branch of the ground state resonance. Parameters as in (a) - solid line, and (c) - dotted line. The inset shows Re u(z) of the unstable perturbation.
Fig. 3
Fig. 3 (a) Instability development of the ground state mode. Parameters as in Fig. 1(a), δ = 3. Spectra of the field across the resonator at the initial (b) and the advanced (c) stages of the comb generation calculated as S ( ε , z ) = | t 0 t 0 + τ ψ ( z , t ) e i ε t d t | : t 0 = 0 in (b), t0 = 1450 in (c) and τ = 50. (d) The spectra at z = 0 and τ = 50: t0 = 0 (blue); t0 = 400 (orange); t0 = 900 (green); t0 = 1400 (red). ϵ in (b)–(d) corresponds to the physical frequency ωp-.
Fig. 4
Fig. 4 (a) Instability development of the n = 8 mode. Parameters as in Fig. 1(a), δ = 5. (b) Spectrum S(ε, z), see Fig. 3, at the advanced stage of the comb generation: t0 = 1450, τ = 50. (c) The spectra at z = 0. The colors and the time intervals same as in Fig. 3(d).

Equations (3)

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i T Ψ = 1 2 d Z 2 Ψ + ( ω 0 + f 2 2 d U ( Z ) ) Ψ i κ 0 Ψ f | Ψ | 2 Ψ f P ( Z ) e i ω p T .
i T ψ = ( 1 / 2 ) Z 2 ψ + ( δ + U ( z ) / 2 ) ψ i κ ψ | ψ | 2 ψ P ( z ) ,
λ u = i ( δ + U ( z ) / 2 ) u + ( i / 2 ) z 2 u κ u + 2 i | ψ ˜ | 2 u + i ψ ˜ 2 v , λ v = i ( δ + U ( z ) / 2 ) v ( i / 2 ) z 2 v κ v 2 i | ψ ˜ | 2 v i ψ ˜ * 2 u ,

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