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High-order analytical formulation of soliton self-frequency shift

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Abstract

We derive an analytical formulation of the Raman-induced frequency shift experienced by a fundamental soliton. By including propagation losses, self-steepening, and dispersion slope, the resulting formulation is a high-order (HO) extension of the well-known Gordon’s formula for soliton self-frequency shift (SSFS). The HO-SSFS formula agrees closely with numerical results of the generalized nonlinear Schrödinger equation, but without the computational complexity and required computation time. The HO-SSFS formula is a useful tool for the design and validation of wavelength conversion systems and supercontinuum generation systems.

© 2021 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2020 (2)

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

I. Alamgir, F. St-Hilaire, and M. Rochette, “All-fiber nonlinear optical wavelength conversion system from the C-band to the mid-infrared,” Opt. Lett. 45, 857–860 (2020).
[Crossref]

2017 (2)

D. D. Hudson, S. Antipov, L. Li, I. Alamgir, T. Hu, M. E. Amraoui, Y. Messaddeq, M. Rochette, S. D. Jackson, and A. Fuerbach, “Toward all-fiber supercontinuum spanning the mid-infrared,” Optica 4, 1163–1166 (2017).
[Crossref]

A. N. Bugay and V. A. Khalyapin, “Analytic description of Raman-Induced frequency shift in the case of non-soliton ultrashort pulses,” Phys. Lett. A 381, 399–403 (2017).
[Crossref]

2014 (1)

2013 (1)

2012 (3)

2010 (1)

2009 (3)

A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. M. de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009).
[Crossref]

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[Crossref]

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

2008 (1)

2007 (2)

G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources (invited),” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
[Crossref]

A. M. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007).
[Crossref]

2005 (2)

K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. 23, 3580–3590 (2005).
[Crossref]

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251, 194–208 (2005).
[Crossref]

2004 (1)

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

2003 (4)

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

K. P. Hansen, “Dispersion flattened hybrid-core nonlinear photonic crystal fiber,” Opt. Express 11, 1503–1509 (2003).
[Crossref]

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003).
[Crossref]

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[Crossref]

2002 (2)

1999 (1)

1995 (1)

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[Crossref]

1993 (1)

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

1992 (1)

J. K. Lucek and K. J. Blow, “Soliton self-frequency shift in telecommunications fiber,” Phys. Rev. A 45, 6666–6674 (1992).
[Crossref]

1989 (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

1986 (1)

1975 (1)

R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
[Crossref]

Agger, C.

Agrawal, G. P.

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[Crossref]

C. J. McKinstrie, J. Santhanam, and G. P. Agrawal, “Gordon–Haus timing jitter in dispersion-managed systems with lumped amplification: analytical approach,” J. Opt. Soc. Am. B 19, 640–649 (2002).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2019).

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Lossless suppression and enhancement of soliton self-frequency shifts,” arXiv:1707.06258 (2017).

Alamgir, I.

I. Alamgir, F. St-Hilaire, and M. Rochette, “All-fiber nonlinear optical wavelength conversion system from the C-band to the mid-infrared,” Opt. Lett. 45, 857–860 (2020).
[Crossref]

D. D. Hudson, S. Antipov, L. Li, I. Alamgir, T. Hu, M. E. Amraoui, Y. Messaddeq, M. Rochette, S. D. Jackson, and A. Fuerbach, “Toward all-fiber supercontinuum spanning the mid-infrared,” Optica 4, 1163–1166 (2017).
[Crossref]

M. H. M. Shamim, I. Alamgir, and M. Rochette, “Efficient supercontinuum generation in As2S3 tapered fiber pumped by soliton-self frequency shifted source,” in OSA Advanced Photonics Congress (AP) 2020 (IPR, NP, NOMA, Networks, PVLED, PSC, SPPCom, SOF) (Optical Society of America, 2020), paper ITh1A.5.

Al-Kadry, A.

Al-Kadry, A. M.

Amraoui, M. E.

Antikainen, A.

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Lossless suppression and enhancement of soliton self-frequency shifts,” arXiv:1707.06258 (2017).

Antipov, S.

Arteaga-Sierra, F. R.

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Lossless suppression and enhancement of soliton self-frequency shifts,” arXiv:1707.06258 (2017).

Atieh, A. K.

Bang, O.

Barviau, B.

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[Crossref]

Biancalana, F.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

Bischel, W. K.

R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
[Crossref]

Blow, K. J.

J. K. Lucek and K. J. Blow, “Soliton self-frequency shift in telecommunications fiber,” Phys. Rev. A 45, 6666–6674 (1992).
[Crossref]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

Brilland, L.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Bugay, A. N.

A. N. Bugay and V. A. Khalyapin, “Analytic description of Raman-Induced frequency shift in the case of non-soliton ultrashort pulses,” Phys. Lett. A 381, 399–403 (2017).
[Crossref]

Caillaud, C.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Chahal, R.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Chen, Z.

Cheremisin, I. I.

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Chrostowski, J.

Coen, S.

Cozic, S.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

de Sterke, C. M.

Dudley, J. M.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources (invited),” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
[Crossref]

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).

Dupont, S.

Efimov, A.

Eggleton, B. J.

Fisher, R. A.

R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
[Crossref]

Freeman, M. J.

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

Fuerbach, A.

Galko, P.

Genty, G.

Ghosh, A. N.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Golant, K. M.

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Gordon, J. P.

Hansen, K. P.

Hasegawa, T.

Hirako, Y.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Hossain, M. A.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Hu, T.

Hudson, D. D.

Huss, G.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Islam, M. A.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Islam, M. N.

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

Jackson, S. D.

Joulain, F.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Judge, A. C.

Kath, W. L.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[Crossref]

Keiding, S. R.

Khalyapin, V. A.

A. N. Bugay and V. A. Khalyapin, “Analytic description of Raman-Induced frequency shift in the case of non-soliton ultrashort pulses,” Phys. Lett. A 381, 399–403 (2017).
[Crossref]

Kibler, B.

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[Crossref]

Kito, C.

Knight, J.

Knight, J. C.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Koshiba, M.

Kozlov, M. V.

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251, 194–208 (2005).
[Crossref]

Kuhlmey, B. T.

Li, L.

Liao, M.

Liu, J.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Lucek, J. K.

J. K. Lucek and K. J. Blow, “Soliton self-frequency shift in telecommunications fiber,” Phys. Rev. A 45, 6666–6674 (1992).
[Crossref]

Lyngsø, J. K.

Mägi, E. C.

Mauricio, J.

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

McKinstrie, C. J.

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251, 194–208 (2005).
[Crossref]

C. J. McKinstrie, J. Santhanam, and G. P. Agrawal, “Gordon–Haus timing jitter in dispersion-managed systems with lumped amplification: analytical approach,” J. Opt. Soc. Am. B 19, 640–649 (2002).
[Crossref]

Meneghetti, M.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Messaddeq, Y.

Miyagi, K.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Miyoshi, S.

Myslinski, P.

Namihira, Y.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Nozaki, S.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Ohishi, Y.

Pant, R.

Perov, A. N.

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Petersen, C.

Picozzi, A.

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[Crossref]

Popov, S. A.

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Poulain, S.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Razzak, S. M. A.

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

Reeves, W.

Roberts, P.

Rochette, M.

I. Alamgir, F. St-Hilaire, and M. Rochette, “All-fiber nonlinear optical wavelength conversion system from the C-band to the mid-infrared,” Opt. Lett. 45, 857–860 (2020).
[Crossref]

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

D. D. Hudson, S. Antipov, L. Li, I. Alamgir, T. Hu, M. E. Amraoui, Y. Messaddeq, M. Rochette, S. D. Jackson, and A. Fuerbach, “Toward all-fiber supercontinuum spanning the mid-infrared,” Optica 4, 1163–1166 (2017).
[Crossref]

A. Al-Kadry, M. E. Amraoui, Y. Messaddeq, and M. Rochette, “Two octaves mid-infrared supercontinuum generation in As2Se3 microwires,” Opt. Express 22, 31131–31137 (2014).
[Crossref]

A. Al-Kadry and M. Rochette, “Maximized soliton self-frequency shift in non-uniform microwires by the control of third-order dispersion perturbation,” J. Lightwave Technol. 31, 1462–1467 (2013).
[Crossref]

A. M. Al-Kadry and M. Rochette, “Mid-infrared sources based on the soliton self-frequency shift,” J. Opt. Soc. Am. B 29, 1347–1355 (2012).
[Crossref]

M. H. M. Shamim, I. Alamgir, and M. Rochette, “Efficient supercontinuum generation in As2S3 tapered fiber pumped by soliton-self frequency shifted source,” in OSA Advanced Photonics Congress (AP) 2020 (IPR, NP, NOMA, Networks, PVLED, PSC, SPPCom, SOF) (Optical Society of America, 2020), paper ITh1A.5.

Rumyantsev, S. D.

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Russell, P. St.J.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

W. Reeves, J. Knight, P. St.J. Russell, and P. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express 10, 609–613 (2002).
[Crossref]

Saitoh, K.

Santhanam, J.

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[Crossref]

C. J. McKinstrie, J. Santhanam, and G. P. Agrawal, “Gordon–Haus timing jitter in dispersion-managed systems with lumped amplification: analytical approach,” J. Opt. Soc. Am. B 19, 640–649 (2002).
[Crossref]

J. Santhanam, “Applications of the moment method to optical communications systems: amplifier noise and timing jitter,” Ph.D. thesis (University of Rochester, 2004).

Sasaoka, E.

Shakhanov, A. V.

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Shamim, M. H. M.

M. H. M. Shamim, I. Alamgir, and M. Rochette, “Efficient supercontinuum generation in As2S3 tapered fiber pumped by soliton-self frequency shifted source,” in OSA Advanced Photonics Congress (AP) 2020 (IPR, NP, NOMA, Networks, PVLED, PSC, SPPCom, SOF) (Optical Society of America, 2020), paper ITh1A.5.

Shebunyaev, A. G.

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Skryabin, D. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Smyth, N. F.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[Crossref]

Steffensen, H.

St-Hilaire, F.

I. Alamgir, F. St-Hilaire, and M. Rochette, “All-fiber nonlinear optical wavelength conversion system from the C-band to the mid-infrared,” Opt. Lett. 45, 857–860 (2020).
[Crossref]

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Suzuki, T.

Sylvestre, T.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Taylor, A. J.

Taylor, J. R.

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).

Terry, F. L.

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

Thøgersen, J.

Thomsen, C. L.

Troles, J.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Venck, S.

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Voronin, A. A.

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

Xia, C.

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

Xie, C.

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251, 194–208 (2005).
[Crossref]

Xu, Z.

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

Yan, X.

Yulin, A. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

Zakel, A.

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

Zheltikov, A. M.

A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008).
[Crossref]

A. M. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

C. Xia, Z. Xu, M. N. Islam, F. L. Terry, M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 µm with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. 15, 422–434 (2009).
[Crossref]

J. Appl. Phys. (1)

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[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (7)

Laser Photon. Rev. (1)

S. Venck, F. St-Hilaire, L. Brilland, A. N. Ghosh, R. Chahal, C. Caillaud, M. Meneghetti, J. Troles, F. Joulain, S. Cozic, S. Poulain, G. Huss, M. Rochette, J. M. Dudley, and T. Sylvestre, “2–10 µm mid-infrared fiber-based supercontinuum laser source: experiment and simulation,” Laser Photon. Rev. 14, 2000011 (2020).
[Crossref]

Opt. Commun. (2)

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251, 194–208 (2005).
[Crossref]

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Optica (1)

Phys. Lett. A (1)

A. N. Bugay and V. A. Khalyapin, “Analytic description of Raman-Induced frequency shift in the case of non-soliton ultrashort pulses,” Phys. Lett. A 381, 399–403 (2017).
[Crossref]

Phys. Rev. A (2)

B. Barviau, B. Kibler, and A. Picozzi, “Wave-turbulence approach of supercontinuum generation: influence of self-steepening and higher-order dispersion,” Phys. Rev. A 79, 063840 (2009).
[Crossref]

J. K. Lucek and K. J. Blow, “Soliton self-frequency shift in telecommunications fiber,” Phys. Rev. A 45, 6666–6674 (1992).
[Crossref]

Phys. Rev. E (3)

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

A. M. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007).
[Crossref]

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[Crossref]

Proc. SPIE (1)

A. V. Shakhanov, K. M. Golant, A. N. Perov, S. D. Rumyantsev, A. G. Shebunyaev, I. I. Cheremisin, and S. A. Popov, “All-silica optical fibers with reduced losses beyond 2 microns,” Proc. SPIE 1893, 85–89 (1993).
[Crossref]

Science (1)

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Other (6)

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Lossless suppression and enhancement of soliton self-frequency shifts,” arXiv:1707.06258 (2017).

M. A. Hossain, Y. Namihira, J. Liu, S. M. A. Razzak, M. A. Islam, Y. Hirako, K. Miyagi, and S. Nozaki, “Dispersion flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” in IET International Conference on Communication Technology and Application (ICCTA) (Institution of Engineering and Technology, 2011), pp. 815–818.

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).

M. H. M. Shamim, I. Alamgir, and M. Rochette, “Efficient supercontinuum generation in As2S3 tapered fiber pumped by soliton-self frequency shifted source,” in OSA Advanced Photonics Congress (AP) 2020 (IPR, NP, NOMA, Networks, PVLED, PSC, SPPCom, SOF) (Optical Society of America, 2020), paper ITh1A.5.

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier, 2019).

J. Santhanam, “Applications of the moment method to optical communications systems: amplifier noise and timing jitter,” Ph.D. thesis (University of Rochester, 2004).

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Figures (8)

Fig. 1.
Fig. 1. Simulation of the (a) temporal and (b) spectral evolution of a fundamental soliton in a 50 m long silica fiber with propagation loss $\alpha = 10\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 77.19\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = 3.1\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 2.08\;{{\rm W}^{- 1}}/{{\rm km}}$ at ${\lambda _0} = 1.94\;\unicode{x00B5}{\rm m}$ . The pulse duration and peak power are 185 fs and 1.08 kW, respectively. The Raman time ${T_R} = 3\;{\rm fs}$ .
Fig. 2.
Fig. 2. Simulation of the evolution of chirp of a fundamental soliton in a 50 m long silica fiber with parameters $\alpha = 10\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 77.19\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = 3.17\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 2.08\;{{\rm W}^{- 1}}/{\rm km}$ at ${\lambda _0} = 1.94\;\unicode{x00B5}{\rm m}$ using moment equations (23)–(27). The pulse duration and peak power are 185 fs and 1.08 kW, respectively.
Fig. 3.
Fig. 3. Simulation of the evolution of pulse duration of a fundamental soliton in a 50 m long silica fiber using the moment method and adiabatic approximation. Fiber parameters are $\alpha = 10\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 77.19\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = 3.17\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 2.08\;{{\rm W}^{- 1}}/{\rm km}$ at ${\lambda _0} = 1.94\;\unicode{x00B5}{\rm m}$ . The pulse duration and peak power are 185 fs and 1.08 kW, respectively.
Fig. 4.
Fig. 4. Evolution of SSFS dominated by self-steepening and propagation loss. The pulse is a fundamental soliton with central wavelength of ${\lambda _0} = 1.55\;\unicode{x00B5}{\rm m}$ , pulse duration of 100 fs, and peak power of 87.4 W. The fiber is a 50 m long dispersion-flattened fiber with $\alpha = 4.2\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 21.85\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = 0\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 25\;{{\rm W}^{- 1}}/{\rm km}$ at ${\lambda _0} = 1.55\;\unicode{x00B5}{\rm m}$ . The inset shows the error of ${\Omega _{\textit{ss}}}$ .
Fig. 5.
Fig. 5. Evolution of SSFS dominated solely by self-steepening in a lossless medium. The pulse is a fundamental soliton with central wavelength of ${\lambda _0} = 1.55\;\unicode{x00B5}{\rm m}$ , pulse duration of 100 fs, and peak power of 87.4 W. The fiber is a 50 m long dispersion-flattened fiber with $\alpha = 0\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 21.85\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = 0\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 25\;{{\rm W}^{- 1}}/{\rm km}$ at ${\lambda _0} = 1.55\;\unicode{x00B5}{\rm m}$ . The inset shows the error of ${\Omega _{\textit{ss}}}$ .
Fig. 6.
Fig. 6. Evolution of SSFS dominated by dispersion slope ( ${\beta _3} \gt 0$ ) and propagation loss. The pulse is a fundamental soliton with central wavelength of ${\lambda _0} = 1.94\;\unicode{x00B5}{\rm m}$ , duration of 185 fs, and peak power of 1.08 kW. The fiber is a 50 m long HNA silica fiber with $\alpha = 10\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 77.19\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = 3.1\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 2.08\;{{\rm W}^{- 1}}/{\rm km}$ at ${\lambda _0} = 1.94\;\unicode{x00B5}{\rm m}$ . In (a), ${\Omega _{{\rm beta}3}}$ is shown with inset that provides the error of ${\Omega _{{\rm beta}3}}$ . In (b), ${\Omega _{\rm{tot}}}$ is shown with inset that provides the error of ${\Omega _{\rm{tot}}}$ .
Fig. 7.
Fig. 7. Evolution of SSFS dominated by dispersion slope ( ${\beta _3} \gt 0$ ) and propagation loss. The pulse is a fundamental soliton with central wavelength of ${\lambda _0} = 1.94\;\unicode{x00B5}{\rm m}$ , pulse duration of 100 fs, and peak power of 886 W. The fiber is a 50 m long ZBLAN fiber with $\alpha = 50\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 18.9\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = 0.112\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 2.1\;{{\rm W}^{- 1}}/{\rm km}$ at ${\lambda _0} = 1.94\;\unicode{x00B5}{\rm m}$ . In (a), ${\Omega _{{\rm beta}3}}$ is shown with inset that provides the error of ${\Omega _{{\rm beta}3}}$ . In (b), ${\Omega _{\rm{tot}}}$ is shown with inset that provides the error of ${\Omega _{\rm{tot}}}$ .
Fig. 8.
Fig. 8. Evolution of SSFS dominated by dispersion slope ( ${\beta _3} \lt 0$ ) in a lossless medium. The fundamental soliton has a central wavelength of ${\lambda _0} = 1.7\;\unicode{x00B5}{\rm m}$ , pulse duration of 100 fs, and peak power of 106 W. The fiber is a 50 m long DF-DDF lossless fiber with $\alpha = 0\;{\rm dB}/{\rm km}$ , ${\beta _2} = - 5.29\;{{\rm ps}^2}/{\rm km}$ , ${\beta _3} = - 0.13\;{{\rm ps}^3}/{\rm km}$ , and ${\gamma _0} = 5\;{{\rm W}^{- 1}}/{\rm km}$ at ${\lambda _0} = 1.7\;\unicode{x00B5}{\rm m}$ . The inset shows the error of ${\Omega _{\rm{tot}}}$ .

Tables (1)

Tables Icon

Table 1. Values of the Coefficient C n

Equations (64)

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Ω G ( z ) = 8 T R | β 2 | 15 T 0 4 z ,
A z + α 2 i n = 1 i n β n n ! n A t n = i γ 0 ( 1 + i ω 0 t ) ( A ( z , t ) 0 R ( t ) | A ( z , t t ) | 2 d t ) ,
R ( t ) = ( 1 f R ) δ ( t ) + f R h R ( t ) ,
T R = 0 t R ( t ) d t f R 0 t h R ( t ) d t .
A z + α 2 A + i β 2 2 2 A T 2 β 3 6 3 A T 3 = i γ 0 ( | A | 2 A + i ω 0 T ( | A | 2 A ) T R A | A | 2 T i T R ω 0 T ( A | A | 2 T ) ) ,
A ( z , T ) = A ( E p , σ p , q p , C p , Ω p , T ) ,
E p ( z ) = | A | 2 d T ,
σ p 2 ( z ) = 1 E p ( T q p ) 2 | A | 2 d T ,
q p ( z ) = 1 E p T | A | 2 d T ,
C p ( z ) = i 2 E p ( T q p ) ( A A T A A T ) d T ,
Ω p ( z ) = i 2 E p ( A A T A A T ) d T .
A ( z , T ) = E p 2 T p s e c h ( T q p T p ) × exp [ i Ω p ( T q p ) i C p ( T q p ) 2 2 T p 2 ] ,
T p 2 = 12 π 2 σ p 2 ,
C p = 12 π 2 C p .
d E p d z = α E p 4 γ 0 T R E p 2 15 ω 0 T p 3 ,
d T p d z = ( β 2 + β 3 Ω p ) C p T p + 4 γ 0 T R E p ω 0 π 2 T p 2 ,
d q p d z = β 2 Ω p + β 3 2 [ Ω p 2 + ( 1 + π 2 C p 2 4 ) 1 3 T p 2 ] + γ 0 E p 2 ω 0 T p ,
d C p d z = ( 4 π 2 + C p 2 ) ( β 2 + β 3 Ω p ) T p 2 + 2 γ 0 ( 1 + Ω p / ω 0 ) E p π 2 T p + ( 150 4 π 2 ) γ 0 T R E p C p 15 ω 0 π 2 T p 3 ,
d Ω p d z = 4 γ 0 ( 1 + Ω p / ω 0 ) T R E p 15 T p 3 + γ 0 C p E p 3 ω 0 T p 3 .
ω 0 , l o c ( z ) = ω 0 + Ω p .
β 2 , l o c ( ω 0 , l o c ) = β 2 ( ω 0 ) + β 3 ( ω 0 ) Ω p ,
γ l o c ( ω 0 , l o c ) = γ ( ω 0 ) + ( γ ( ω 0 ) / ω 0 ) Ω p .
d E p d z = α E p 4 γ 0 T R E p 2 15 ω 0 T p 3 ,
d T p d z = β 2 , l o c C p T p + 4 γ 0 T R E p ω 0 π 2 T p 2 ,
d q p d z = β 3 6 T p 2 ( 1 + π 2 C p 2 4 ) + γ 0 E p 2 ω 0 T p ,
d C p d z = ( 4 π 2 + C p 2 ) β 2 , l o c T p 2 + 2 γ l o c E p π 2 T p + ( 150 4 π 2 ) γ 0 T R E p C p 15 ω 0 π 2 T p 3 ,
d Ω p d z = 4 γ l o c T R E p 15 T p 3 + γ 0 C p E p 3 ω 0 T p 3 .
N 2 = γ l o c E p T p 2 | β 2 , l o c | = 1 ,
T p ( z ) = 2 | β 2 , l o c | γ l o c E p .
d C p d z = γ l o c 2 E p 2 C p 2 4 | β 2 , l o c | + ( 150 4 π 2 ) γ 0 γ l o c 3 T R E p 4 C p 120 ω 0 π 2 | β 2 , l o c | 3 .
d Ω p d z = 4 γ l o c T R E p 15 T p 3 .
d Ω p d z = γ l o c 4 T R E p 4 30 | β 2 , l o c | 3 .
0 Ω p | β 2 ( ω 0 ) + β 3 ( ω 0 ) Ω p | 3 ( γ ( ω 0 ) + γ ( ω 0 ) ω 0 Ω p ) 4 d Ω p = T R 30 0 z E p 4 d z .
a Ω p 4 + b Ω p 3 + c Ω p 2 + d Ω p + e = 0 ,
a = 6 γ 0 6 β 3 3 ω 0 4 ,
b = γ 0 6 β 2 3 ω 0 6 ( 2 + 3 ω 0 β 3 β 2 + 6 ω 0 2 β 3 2 β 2 2 + 7 ω 0 3 β 3 3 β 2 3 γ 0 4 T R E p 0 4 z e f f 5 ω 0 β 2 3 ) ,
c = 3 γ 0 6 β 2 3 ω 0 5 ( 2 + 3 ω 0 β 3 β 2 + ω 0 3 β 3 3 β 2 3 γ 0 4 T R E p 0 4 z e f f 5 ω 0 β 2 3 ) ,
d = 3 γ 0 6 β 2 3 ω 0 4 ( 2 γ 0 4 T R E p 0 4 z e f f 5 ω 0 β 2 3 ) ,
e = γ 0 10 T R E p 0 4 z e f f 5 ω 0 4 ,
z e f f = 1 4 α ( 1 e 4 α z ) ,
a Ω p 3 + b Ω p 2 + c Ω p + d = 0 ,
a = γ 0 3 ω 0 3 ( 8 T R | β 2 | z e f f 5 ω 0 T 0 4 + 1 ) ,
b = 3 γ 0 3 ω 0 2 ( 8 T R | β 2 | z e f f 5 ω 0 T 0 4 + 1 ) ,
c = 3 γ 0 3 ω 0 ( 8 T R | β 2 | z e f f 5 ω 0 T 0 4 + 1 ) ,
d = 8 γ 0 3 T R | β 2 | z e f f 5 ω 0 T 0 4 .
( Ω p ω 0 + 1 ) 3 = 5 ω 0 T 0 4 ( 8 T R | β 2 | z e f f + 5 ω 0 T 0 4 ) .
Ω ss ( z ) = ω 0 [ 1 ( 5 ω 0 T 0 4 8 T R | β 2 | z e f f + 5 ω 0 T 0 4 ) 3 ] .
Ω ss ( z ) = 8 T R | β 2 | 15 T 0 4 z e f f + n = 2 ( 1 ) n 2 3 n T R n | β 2 | n 3 n 1 n ! 15 n ω 0 n 1 T 0 4 n × Γ ( 1 3 + n ) Γ ( 4 3 ) z e f f n ,
Ω G ( z ) = 8 T R | β 2 | 15 T 0 4 z e f f .
Δ Ω ss = Ω ss Ω G ,
Δ Ω ss = n = 2 ( 1 ) n 2 3 n T R n | β 2 | n 3 n 1 n ! 15 n ω 0 n 1 T 0 4 n Γ ( 1 3 + n ) Γ ( 4 3 ) z e f f n .
0 Ω p | β 2 ( ω 0 ) | 4 | β 2 ( ω 0 ) + β 3 ( ω 0 ) Ω p | 3 γ 0 4 Ω p 4 ( | β 2 ( ω ) | Ω p + s g n ( β 3 ) | β 3 ( ω 0 ) | + | β 2 ( ω 0 ) | ω 0 ) 4 d Ω p = T R 30 0 z E p 4 d z ,
0 Ω p | β 2 ( ω 0 ) + β 3 ( ω 0 ) Ω p | 3 γ 0 4 d Ω p = T R 30 0 z E p 4 d z .
Ω b e t a 3 = | β 2 | β 3 [ ( 1 + 32 T R β 3 15 T 0 4 z e f f ) 4 1 ] .
Ω b e t a 3 = 8 T R | β 2 | 15 T 0 4 z e f f + n = 2 [ ( 1 ) n 2 3 n T R n | β 2 | β 3 n 1 4 n 1 n ! 15 n T 0 4 n Γ ( n 1 4 ) Γ ( 3 4 ) ] z e f f n .
Δ Ω b e t a 3 = Ω b e t a 3 Ω G ,
Δ Ω b e t a 3 = n = 2 [ ( 1 ) n 2 3 n T R n | β 2 | β 3 n 1 4 n 1 n ! 15 n T 0 4 n Γ ( n 1 4 ) Γ ( 3 4 ) ] z e f f n .
Ω t o t = Ω G + Δ Ω ss + Δ Ω b e t a 3 + O ( z e f f 3 ) ,
Ω t o t = ω 0 [ 1 ( 5 ω 0 T 0 4 8 T R | β 2 | z e f f + 5 ω 0 T 0 4 ) 3 ] | β 2 | β 3 [ ( 1 + 32 T R β 3 15 T 0 4 z e f f ) 4 1 ] + 8 T R | β 2 | 15 T 0 4 z e f f .
| Δ Ω b e t a 3 , n | | Δ Ω s s , n | = B n [ | β 3 n 1 | | β 2 | n 1 ω 0 n 1 C n ] | z e f f n | ,
B n = 2 5 n 2 T R n | β 2 | n ! 15 n T 0 4 n Γ ( n 1 4 ) Γ ( 3 4 ) ,
C n = ( 3 4 ) n 1 Γ ( 1 3 + n ) Γ ( 3 4 ) Γ ( 4 3 ) Γ ( n 1 4 ) .
Ω G N L S E ( z ) = i = 1 k ω i | A ~ ( z , ω i ) | 2 i = 1 k | A ~ ( z , ω i ) | 2 ,
E r r o r ( % ) = | Ω G N L S E Ω | | Ω G N L S E | × 100.

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