Abstract

We rigorously analyze nonlinear propagation in hybrid silicon-plasmonic (HSP) waveguides. Focusing on the relative importance of Kerr and free carrier effects (FCE) originating from two-photon absorption, we establish a set of figures of merit applicable to any silicon-comprising waveguide. An optimized HSP design is proposed, deeply confining the optical field in a nanosized nonlinear polymer gap formed between a metal wedge and an underlying silicon wire. An exceptionally high nonlinear parameter γ>104m1W1 is attained, combined with an FCE power threshold larger than 1 W, even in CW. The formulation is also extended to multimode waveguides, supported by two all-optical applications of an HSP nonlinear directional coupler.

© 2013 Optical Society of America

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2012

2011

2010

2009

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

S. Afshar Vahid and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009).
[CrossRef]

2008

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement, and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

A. Boltasseva, V. S. Volkov, R. B. Nielsen, E. Moreno, S. G. Rodrigo, and S. I. Bozhevolnyi, “Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths,” Opt. Express 16, 5252–5260 (2008).
[CrossRef]

2007

2006

2005

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

2003

A. Zakery and S. R. Elliott, “Optical properties, and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[CrossRef]

1994

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

1987

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

1982

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18, 1580–1583 (1982).
[CrossRef]

Afshar Vahid, S.

Agrawal, G. P.

Baets, R.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Baker, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Berkovitch, N.

Biaggio, I.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Bian, Y.

Bogaerts, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Boltasseva, A.

Bozhevolnyi, S. I.

Butcher, P.

P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990).

Cha, M.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Chen, X.

Chetrit, Y.

Ciftcioglu, B.

Claps, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Cotter, D.

P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University, 1990).

Dai, D.

Daniel, B. A.

Diederich, F.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Dimitropoulos, D.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Dulkeith, E.

Dumon, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Elliott, S. R.

A. Zakery and S. R. Elliott, “Optical properties, and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[CrossRef]

Esembeson, B.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Etemad, S.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Fathpour, S.

Freude, W.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement, and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Ginzburg, P.

Gu, M.

Han, Z.

Hayat, A.

He, S.

Hossain, M. M.

Izhaky, N.

Jalali, B.

B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006).
[CrossRef]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. 18, 1580–1583 (1982).
[CrossRef]

Jhaveri, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Kang, J. U.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Koos, C.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Krasavin, A. V.

Kriezis, E. E.

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis, and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON-2012: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.

Lawrence, B. L.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Leuthold, J.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Li, Z.

Liao, L.

Lin, Q.

Liu, A.

Liu, J.

Liu, Y.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Meng, Z.

Meth, J.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Michinobu, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Milian, C.

C. Milian and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

Monro, T. M.

Moreno, E.

Nguyen, H.

Nielsen, R. B.

Orenstein, M.

Osgood, R. M.

Oulton, R. F.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement, and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Painter, O. J.

Paniccia, M.

Panoiu, N. C.

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement, and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Pitilakis, A.

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis, and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON-2012: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.

Premaratne, M.

Qin, F.

Rodrigo, S. G.

Rubin, D.

Rukhlenko, I. D.

Scimeca, M. L.

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Shi, Y.

Skryabin, D. V.

C. Milian and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Soref, R. A.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement, and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Stegeman, G.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Tasolamprou, A. C.

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis, and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

Thylen, L.

Toruellas, W.

B. L. Lawrence, M. Cha, J. U. Kang, W. Toruellas, G. Stegeman, G. Baker, J. Meth, and S. Etemad, “Large purely refractive nonlinear index of single crystal p-toluene sulphonate (pts) at 1600 nm,” Electron. Lett. 30, 447–448 (1994).
[CrossRef]

Tsilipakos, O.

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis, and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, “Nonlinear effects in hybrid plasmonic waveguides,” in ICTON-2012: 14th International Conference on Transparent Optical Networks (IEEE, 2012), paper 6254436.

Turner, M. D.

Vallaitis, T.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Vien, V.

Vlasov, Y. A.

Volkov, V. S.

Vorreau, P.

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 3, 216–219 (2009).
[CrossRef]

Woo, J. C. S.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[CrossRef]

Wosinski, L.

Wu, M.

Yin, L.

Yioultsis, T. V.

O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou, T. V. Yioultsis, and E. E. Kriezis, “Computational techniques for the analysis, and design of dielectric-loaded plasmonic circuitry,” Opt. Quantum Electron. 42, 541–555 (2011).
[CrossRef]

Zakery, A.

A. Zakery and S. R. Elliott, “Optical properties, and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1–12 (2003).
[CrossRef]

Zayats, A. V.

Zhang, X.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement, and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[CrossRef]

Zheng, Z.

Zhong, X.

Zhou, T.

Zhu, J.

Adv. Mater.

B. Esembeson, M. L. Scimeca, T. Michinobu, F. Diederich, and I. Biaggio, “A high-optical quality supramolecular assembly for third-order integrated nonlinear optics,” Adv. Mater. 20, 4584–4587 (2008).
[CrossRef]

Appl. Phys. Lett.

C. Milian and D. V. Skryabin, “Nonlinear switching in arrays of semiconductor on metal photonic wires,” Appl. Phys. Lett. 98, 111104 (2011).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Typical guided-power density distribution, Pz(x,y), for three waveguide types: (a) TE mode of a silicon wire of W×H=300nm×200nm, (b) TE mode of a 50 nm-wide low-index slot formed between two W×H=125nm×200nm Si wires, (c) TM mode of a generic HSP waveguide formed by a 50 nm gap separating a metal layer and a W×H=300nm×150nm Si wire. In all three waveguides, the cladding is a low-index dielectric with n0=1.8.

Fig. 2.
Fig. 2.

Cross section of the hybrid silicon-plasmonic (HSP) waveguides: (a) standard CGS, (b) extended CGS with inverted metal rib, and (c) extended CGS with inverted metal wedge.

Fig. 3.
Fig. 3.

Effect of gap size in the extended CGS waveguide with W×H=220nm×340nm, tm=50nm and tr=0. (a) Nonlinear parameter and propagation length, (b) TPA ratio and carrier field overlap factor, (c) effective index and longitudinal enhancement factor.

Fig. 4.
Fig. 4.

Effect of gap size on figures of merit F and ζfc, for the extended CGS waveguide with W×H=220nm×340nm, tm=50nm and tr=0.

Fig. 5.
Fig. 5.

Parametric investigation of the inverted metal wedge HSP waveguide with respect to (a) the wedge tip curvature radius Rw, (b) the wedge angle φw, and (c) the wedge thickness tw. In all cases: W×H=220nm×220nm, tm=50nm and g=20nm. When not parametrically varied: Rw=1nm, φw=2tan1(0.5)53.2° and tw=100nm.

Fig. 6.
Fig. 6.

Normalized intensity distribution of the inverted metal wedge HSP waveguide TM00 mode. (a) Reference design with Rw=1nm and φw=53.2°, (b) larger wedge angle φw=102.7°, and (c) larger tip curvature Rw=10nm. In all cases: W×H=220nm×220nm, tm=50nm and g=20nm.

Fig. 7.
Fig. 7.

NLSE-calculated (a) nonlinear insertion losses ILNL and (b) nonlinear phase shift fraction rδ as a function of input power. The dotted lines in each subfigure correspond to the threshold powers Pth,FCA=(α/fA)1/2 and Pth,FCD=γ/fD, respectively, for each waveguide. The waveguide parameters are given in Table 1.

Fig. 8.
Fig. 8.

Electric field distributions, Re{ey(x,y)}, of the (a) symmetric/even and (b) anti-symmetric/odd TM-like supermodes of a directional coupler formed by a pair of HSP waveguides.

Fig. 9.
Fig. 9.

(a) CW characterization of the NLDC, i.e., XT between the output waveguides as a function of the total input power, for both TM (solid lines) and TE (dotted lines) polarizations. The three HSP waveguide spacings, dw=600, 700, 800 nm, correspond to lengths L=LcTM=14.6, 28.2, 54.0 μm, respectively. (b) Extinction ratio increases at the output of the left waveguide of the NLDC when a TM-polarized 10 GHz NRZ-modulated signal of ER=5dB is launched in its left input-waveguide. The peak power is 14 dBW and the NLDC is 28.2 μm long (dw=700nm).

Tables (1)

Tables Icon

Table 1. Nonlinear Waveguide NLSE Parameters

Equations (30)

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F=γLprop=γ/α,
Az=[i(γ|A|2fD|A|4)12(α+2rγ|A|2+2fA|A|4)]A,
F=γα=F1(fD/γ)|A|21+2rF|A|2+2(fA/α)|A|4.
γc=3ω0ε04N2μ,α,β,γx,y,zχμαβγ(3)eμ*eαeβ*eγdxdy,
χμαβγ(3)=χ1111(3)[ρ3(δμαδβγ+δμβδαγ+δμγδαβ)+(1ρ)qRqμRqαRqβRqγ],
χ1111(3)=43ε0cn02n2(1+iβTPA2k0n2),
Nt=GNτfc,
N(z,t)=IFT{FT{G(z,t)}τfc1iω},
N(x,y,z)=PTPA/z2ω0τfc=βTPAP˜z2|A|42ω0τfc,
Δufc=Δufc(x,y,z)|e|2dxdy|e|2dxdy=σuN,
N=Ξ2ω0|A|4τfc,
Ξ=βTPAP˜z2=βTPAP˜z2|e|2dxdy|e|2dxdy
fA|A|4=+ΓnSineff×12Δαfc,
fD|A|4=ΓnSineff×k0Δnfc,
Γ=c0ε0neffN1|e|2dxdy.
fA=ΓnSineff×Ξτfc2ω0×12σα,
fD=ΓnSineff×Ξτfc2ω0×k0σn.
ζfc=γneffΞΓ,
rδ=Φnon-KerrΦKerr=γPinLeffΔΦNLγPinLeff0,
ILNL=|A|2Pinexp(L/Lprop).
Akz=[n=1in+1βn(k)n!ntn]Akαk2Ak+T3ok+Tfck.
T3ok(z,t)=l,m,n=1KiγklmnAlAm*Anexp(iΔβklmnz),
Tfck(z,t)=l=1KiδklAlexp{i[β0(l)β0(m)]z},
γklmn=3ω0ε04μαβγxyzχμαβγ(3)eμ*(k)eα(l)eβ*(m)eγ(n)(NkNlNmNn)1/2dxdy,
δkl(z,t)=ω0ε0nSi(NkNl)1/2Δufc[e*(k)·e(l)]dxdy,
δkk(z,t)=k0nSiΓ(k)neff(k)σuNk,
G(x,y,z,t)=βTPA2ω0(m=1KP˜z(m)|Am|2)2.
Nk=τfcGk=τfc2ω0m,n=1K|Am|2|An|2Ξmn(k),
Ξmn(k)=βTPAP˜z(m)P˜z(n)k
Psw>π3F[1exp(Lc/Lprop)].

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