Abstract

Modeling of the soliton self-frequency shift (SSFS) using the generalized nonlinear Schrödinger equation (GNLSE) is computationally intensive and becomes time consuming, particularly when comparing the self-frequency shift of several fibers. We present a simple theory that combines fission of a higher-order soliton with the Gordon equation for the evolution of a fundamental soliton. The theory allows the computation of the final soliton wavelength using a simple integration that is far less time consuming than solving the full GNLSE. In the simplest version of the theory no integration is even required. This approach was applied to compare the SSFS in photonic crystal fibers with different dispersion and nonlinear parameters, and the fiber that exhibited the maximum SSFS was selected. Findings of the theory were confirmed with full-scale simulations of the modified GNLSE and with experiments showing that the theoretically predicted fiber optimizes the self-frequency shift.

© 2010 Optical Society of America

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2009 (1)

2008 (3)

2007 (1)

2006 (3)

2004 (2)

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748–751 (2004).

M. Kato, K. Fujiura, and T. Kurihara, “Asynchronous all-optical bit-by-bit self-signal recognition and demultiplexing from overlapped signals achieved by self-frequency shift of Raman soliton,” Electron. Lett. 40, 381–382 (2004).
[CrossRef]

2003 (2)

C. Xu and X. Liu, “Photonic analog-to-digital converter using soliton self-frequency shift and interleaving spectral filters,” Opt. Lett. 28, 986–988 (2003).
[CrossRef] [PubMed]

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

2002 (2)

I. G. Cormack, D. T. Reid, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, “Observation of soliton self-frequency shift in photonic crystal fiber,” Electron. Lett. 38, 167–169 (2002).
[CrossRef]

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt. Express 10, 1083–1098 (2002).
[PubMed]

2001 (3)

X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windler, “Soliton self-frequency shift in a short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358–360 (2001).
[CrossRef]

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

1997 (1)

H. Hatami-Hanza, J. Hong, A. Atieh, P. Myslinski, and J. Chrostowski, “Demonstration of all-optical demultiplexing of a multilevel soliton signal employing soliton decomposition and self-frequency shift,” IEEE Photon. Technol. Lett. 9, 833–835 (1997).
[CrossRef]

1994 (1)

1992 (1)

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

1986 (2)

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Atieh, A.

H. Hatami-Hanza, J. Hong, A. Atieh, P. Myslinski, and J. Chrostowski, “Demonstration of all-optical demultiplexing of a multilevel soliton signal employing soliton decomposition and self-frequency shift,” IEEE Photon. Technol. Lett. 9, 833–835 (1997).
[CrossRef]

Baltuska, A.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Banaee, M. G.

Bang, O.

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] [PubMed]

Broeng, J.

Brownell, M.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Chandalia, J. K.

Chrostowski, J.

H. Hatami-Hanza, J. Hong, A. Atieh, P. Myslinski, and J. Chrostowski, “Demonstration of all-optical demultiplexing of a multilevel soliton signal employing soliton decomposition and self-frequency shift,” IEEE Photon. Technol. Lett. 9, 833–835 (1997).
[CrossRef]

Coen, S.

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Cormack, I. G.

I. G. Cormack, D. T. Reid, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, “Observation of soliton self-frequency shift in photonic crystal fiber,” Electron. Lett. 38, 167–169 (2002).
[CrossRef]

Crosby, P. A.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

de Sterke, C. M.

Dudley, J. M.

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Eggleton, B. J.

Fateev, N. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748–751 (2004).

Fuji, T.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Fujiura, K.

M. Kato, K. Fujiura, and T. Kurihara, “Asynchronous all-optical bit-by-bit self-signal recognition and demultiplexing from overlapped signals achieved by self-frequency shift of Raman soliton,” Electron. Lett. 40, 381–382 (2004).
[CrossRef]

Genty, G.

Golding, P. S.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Gordon, J. P.

Grange, R.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Haiml, M.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Hatami-Hanza, H.

H. Hatami-Hanza, J. Hong, A. Atieh, P. Myslinski, and J. Chrostowski, “Demonstration of all-optical demultiplexing of a multilevel soliton signal employing soliton decomposition and self-frequency shift,” IEEE Photon. Technol. Lett. 9, 833–835 (1997).
[CrossRef]

Herrmann, J.

Hong, J.

H. Hatami-Hanza, J. Hong, A. Atieh, P. Myslinski, and J. Chrostowski, “Demonstration of all-optical demultiplexing of a multilevel soliton signal employing soliton decomposition and self-frequency shift,” IEEE Photon. Technol. Lett. 9, 833–835 (1997).
[CrossRef]

Howe, J. V.

J. H. Lee, J. V. Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Quantum Electron. 14, 713–723 (2008).
[CrossRef]

Ishii, N.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Judge, A.

Kaivola, M.

Kato, M.

M. Kato, K. Fujiura, and T. Kurihara, “Asynchronous all-optical bit-by-bit self-signal recognition and demultiplexing from overlapped signals achieved by self-frequency shift of Raman soliton,” Electron. Lett. 40, 381–382 (2004).
[CrossRef]

Keller, U.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Kilburn, I. J.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Knight, J. C.

I. G. Cormack, D. T. Reid, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, “Observation of soliton self-frequency shift in photonic crystal fiber,” Electron. Lett. 38, 167–169 (2002).
[CrossRef]

Knox, W. H.

Kobtsev, S. M.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748–751 (2004).

Kohler, S.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Kosinski, S. G.

Krainer, L.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Krausz, F.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Kuhlmey, B. T.

Kukarin, S. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748–751 (2004).

Kurihara, T.

M. Kato, K. Fujiura, and T. Kurihara, “Asynchronous all-optical bit-by-bit self-signal recognition and demultiplexing from overlapped signals achieved by self-frequency shift of Raman soliton,” Electron. Lett. 40, 381–382 (2004).
[CrossRef]

Lacourt, P. A.

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Lægsgaard, J.

Lee, J. H.

J. H. Lee, J. V. Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Quantum Electron. 14, 713–723 (2008).
[CrossRef]

Lehtonen, M.

Liu, X.

Liu, Y. -g.

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] [PubMed]

Ludvigsen, H.

Magi, E. C.

Maruta, A.

Metzger, T.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Mitschke, F.

Mollenauer, L. F.

Myslinski, P.

H. Hatami-Hanza, J. Hong, A. Atieh, P. Myslinski, and J. Chrostowski, “Demonstration of all-optical demultiplexing of a multilevel soliton signal employing soliton decomposition and self-frequency shift,” IEEE Photon. Technol. Lett. 9, 833–835 (1997).
[CrossRef]

Nazarkin, A.

Oda, S.

Pant, R.

Paschotta, R.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Ralph, S. E.

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Reid, D. T.

I. G. Cormack, D. T. Reid, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, “Observation of soliton self-frequency shift in photonic crystal fiber,” Electron. Lett. 38, 167–169 (2002).
[CrossRef]

Rhodes, W. T.

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Russell, P. S. J.

I. G. Cormack, D. T. Reid, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, “Observation of soliton self-frequency shift in photonic crystal fiber,” Electron. Lett. 38, 167–169 (2002).
[CrossRef]

Serebryannikov, E. E.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Smirnov, S. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748–751 (2004).

Spuhler, G. J.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Teisset, C. Y.

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Voronin, A. A.

Wadsworth, W. J.

I. G. Cormack, D. T. Reid, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, “Observation of soliton self-frequency shift in photonic crystal fiber,” Electron. Lett. 38, 167–169 (2002).
[CrossRef]

Wang, D.

Washburn, B. R.

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Weingarten, K. J.

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

Windeler, R. S.

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Windler, R. S.

Xu, C.

Young, J. F.

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

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] [PubMed]

N. Ishii, C.Y. Teisset, S. Kohler, E. E. Serebryannikov, T. Fuji, T. Metzger, F. Krausz, A. Baltuska, and A. M. Zheltikov, “Widely tunable soliton frequency shifting of few-cycle laser pulses,” Phys. Rev. E 74, 036617 (2006).
[CrossRef]

Electron. Lett. (4)

M. Kato, K. Fujiura, and T. Kurihara, “Asynchronous all-optical bit-by-bit self-signal recognition and demultiplexing from overlapped signals achieved by self-frequency shift of Raman soliton,” Electron. Lett. 40, 381–382 (2004).
[CrossRef]

G. J. Spuhler, P. S. Golding, L. Krainer, I. J. Kilburn, P. A. Crosby, M. Brownell, K. J. Weingarten, R. Paschotta, M. Haiml, R. Grange, and U. Keller, “Multi-wavelength source with 25 GHz channel spacing tunable over C-band,” Electron. Lett. 39, 778–780 (2003).
[CrossRef]

B. R. Washburn, S. E. Ralph, P. A. Lacourt, J. M. Dudley, W. T. Rhodes, R. S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibers,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

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[CrossRef]

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J. H. Lee, J. V. Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Quantum Electron. 14, 713–723 (2008).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic of the experimental setup (left) for studying SSFS based wavelength tuning using variation in pump power (right).

Fig. 2
Fig. 2

R ( T ) in silica versus the pulse width ( T ) of the fundamental soliton.

Fig. 3
Fig. 3

(a) Shift rate and (b) energy in the largest soliton for the three commercial fibers versus wavelength for P 0 = 6.5   kW . (c) Illustration of limited analysis in Subsection 2D with the β 2 ratios for commercial fibers shown using symbols and dotted and dashed-dotted lines.

Fig. 4
Fig. 4

Wavelength shift of the output soliton, obtained by integrating Eq. (8), as a function of the input soliton power for NL-15-670, NL-17-700, and NL-18-710 PCFs in (a) 50 cm long PCFs at λ P = 785   nm and (b) 45 cm long PCFs at λ P = 806   nm .

Fig. 5
Fig. 5

Comparison of (a) theoretical (solid line) and GNLSE simulated (circles) wavelength shifts at pump wavelength of 806 nm using L = 45   cm , (b) theoretical (solid line) and experimental (squares) wavelength shifts at pump wavelength of 806 nm, (c) theoretical (solid line) and GNLSE simulated (circles) wavelength shifts at the pump wavelength of 785 nm using L = 50   cm , and (d) theoretical (solid line) and experimental (squares) wavelength shifts at pump wavelength of 785 nm using L = 50   cm for all the fibers.

Fig. 6
Fig. 6

(a)–(d) Comparison of the wavelength shifts using the simple theory (vertical line), and experimental (solid line) and simulated (dashed line) SSFS spectra for NL-15-670, for different input powers using λ p = 785   nm , L = 50   cm .

Equations (14)

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T k ( ω P ) = T 0 2 N + 1 2 k ,
P k ( ω P ) = ( 2 N + 1 2 k ) 2 N 2 P 0 ,     k = 1 , , N ,
T k ( ω ¯ ) = 2 | β 2 ( ω ¯ ) | γ ( ω ¯ ) E k ( ω ¯ ) ,
ω ¯ z = | β 2 ( ω ¯ ) | T k 3 ( ω ¯ ) R ( T k ( ω ¯ ) ) ,
R ( T k ( ω ¯ ) ) = f R π 2 T k 4 ( ω ¯ ) 6 α R ( ω ) ω 3 sinh 2 ( π ω T k ( ω ¯ ) / 2 ) d ω 2 π ,
ω ¯ z = γ 3 ( ω ¯ ) E k 3 ( ω ¯ ) 8 | β 2 ( ω ¯ ) | 2 R ( T k ( ω ¯ ) ) .
E k = 2 P k T k = 2 P 0 T 0 ( 2 N + 1 2 k ) N 2 .
E 1 ( ω P ) = 2 P 1 T 1 = 2 P 0 T 0 ( 2 N 1 ) N 2 2 N E 0 4 P 0 | β 2 ( ω P ) | γ ( ω P ) .
ω ¯ z = 8 γ 3 ( ω ¯ ) ω 3 ¯ | β 2 ( ω ¯ ) | 2 ω P 3 P 0 3 | β 2 ( ω P ) | 3 γ 3 ( ω P ) R ( T 1 ( ω ¯ ) ) .
ω ¯ z = γ 3 ( ω P ) P 0 3 | β 2 ( ω P ) | R ( T 1 ( ω P ) ) ,
T 1 ( ω P ) = 1 2 β 2 ( ω P ) γ ( ω P ) P 0 .
γ A 3 R 2 ( T A ) | β 2 A | > γ B 3 R 2 ( T B ) | β 2 B | ,     | β 2 A | γ A > | β 2 B | γ B ,
γ A γ B < | β 2 ( A ) | | β 2 ( B ) | γ A 3 γ B 3 .
| β 2 ( A ) | | β 2 ( B ) | < γ A γ B ,     | β 2 ( A ) | | β 2 ( B ) | γ A 3 γ B 3 ,

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