Abstract

In this paper a method is proposed to maximize the bandwidth of the WDM router based on one-pump fiber parametric wavelength converters. It is proved that for such converters there exists an optimum signal (idler) frequency at which the output (input) tuning range can be maximized. Analytical expressions of the optimum frequency and the maximal tuning range are deduced. Then a two-stage bidirectional wavelength conversion method is proposed. With this method the bandwidth of the WDM router based on such a converter can be significantly improved compared to the one-stage ones by 252% if ordinary highly nonlinear fibers are used or 390% if fibers with optimal fourth order dispersion are used.

© 2009 Optical Society of America

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  6. P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
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  13. T. Yamamoto and M. Nakazawa, "Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching," IEEE Photonics Tech. Lett. 9, 327-329 (1997).
    [CrossRef]
  14. G. Kalogerakis and M. E. Marhic, "Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters," J. Lightwave Tech. 24, 3683-3691 (2006).
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    [CrossRef]
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    [CrossRef]
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2008 (3)

2007 (1)

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

2006 (2)

G. Kalogerakis and M. E. Marhic, "Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters," J. Lightwave Tech. 24, 3683-3691 (2006).
[CrossRef]

F. Yaman, Q. Lin, and G. P. Agrawal, "A novel design for polarization-independent single-pump fiber-optic parametric amplifier," IEEE Photonics Tech. Lett. 18, 2335-2337 (2006).
[CrossRef]

2005 (4)

T. Yang, C. Shu and C. Lin, "Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber," Optics Express 13, 5409-5415 (2005).
[CrossRef] [PubMed]

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
[CrossRef]

K. K. Chow, C. Shu, C. Lin, and A. Bjarklev, "Polarization-Insensitive Widely Tunable Wavelength Converter Based on Four-Wave Mixing in a Dispersion-Flattened Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 624-626 (2005).
[CrossRef]

A. Zhang and M. S. Demokan, "Broadband wavelength converter based on four-wave mixing in a highly nonlinear photonic crystal fiber," Opt. Lett. 30,2375-2376 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

2002 (1)

M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, "Transparent wavelength conversion in fibre with 24nm pump tuning range," Electron. Lett. 38, 85-86 (2002).
[CrossRef]

2001 (1)

J. Hansryd and P. A. Andrekson, "Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency," IEEE Photon. Tech. Lett. 13,194-196 (2001).
[CrossRef]

1999 (1)

M. E. Marhic, F. S. Yang, Min-Chen Ho and L. G. kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Lightwave Tech. 17, 210-215 (1999).
[CrossRef]

1998 (1)

1997 (1)

T. Yamamoto and M. Nakazawa, "Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching," IEEE Photonics Tech. Lett. 9, 327-329 (1997).
[CrossRef]

1996 (1)

1995 (1)

M. Yu and C. J. McKinstrie, "Modulational instabilities in dispersion-flattened fibers," Physical Review E,  52, 1072-1080 (1995).
[CrossRef]

1994 (1)

Aggarwal, I. D.

Agrawal, G. P.

F. Yaman, Q. Lin, and G. P. Agrawal, "A novel design for polarization-independent single-pump fiber-optic parametric amplifier," IEEE Photonics Tech. Lett. 18, 2335-2337 (2006).
[CrossRef]

Andersen, P. A.

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
[CrossRef]

Andrekson, P. A.

M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, "Transparent wavelength conversion in fibre with 24nm pump tuning range," Electron. Lett. 38, 85-86 (2002).
[CrossRef]

J. Hansryd and P. A. Andrekson, "Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency," IEEE Photon. Tech. Lett. 13,194-196 (2001).
[CrossRef]

Bjarklev, A.

K. K. Chow, C. Shu, C. Lin, and A. Bjarklev, "Polarization-Insensitive Widely Tunable Wavelength Converter Based on Four-Wave Mixing in a Dispersion-Flattened Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 624-626 (2005).
[CrossRef]

Blows, J. L.

Chiang, T. K.

Chow, K. K.

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

K. K. Chow, C. Shu, C. Lin, and A. Bjarklev, "Polarization-Insensitive Widely Tunable Wavelength Converter Based on Four-Wave Mixing in a Dispersion-Flattened Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 624-626 (2005).
[CrossRef]

de Sterke, C. M

de Sterke, M.

Demokan, M. S.

Eggleton, B. J.

Farahmand, M.

Fu, L.

Geng, Y.

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
[CrossRef]

Hansryd, H.

M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, "Transparent wavelength conversion in fibre with 24nm pump tuning range," Electron. Lett. 38, 85-86 (2002).
[CrossRef]

Hansryd, J.

J. Hansryd and P. A. Andrekson, "Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency," IEEE Photon. Tech. Lett. 13,194-196 (2001).
[CrossRef]

Hasegawa, T.

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

Ho, Min-Chen

M. E. Marhic, F. S. Yang, Min-Chen Ho and L. G. kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Lightwave Tech. 17, 210-215 (1999).
[CrossRef]

Inoue, K.

Jeppesen, P.

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
[CrossRef]

Kagi, N.

Kalogerakis, G.

G. Kalogerakis and M. E. Marhic, "Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters," J. Lightwave Tech. 24, 3683-3691 (2006).
[CrossRef]

Karlsson, M.

kazovsky, L. G.

M. E. Marhic, F. S. Yang, Min-Chen Ho and L. G. kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Lightwave Tech. 17, 210-215 (1999).
[CrossRef]

M. E. Marhic, N. Kagi, T. K. Chiang, and L. G. Kazovsky, "Broadband fiber optical parametric amplifiers," Opt. Lett. 21, 573-575 (1996).
[CrossRef] [PubMed]

Kikuchi, K.

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

Knudsen, S. N.

M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, "Transparent wavelength conversion in fibre with 24nm pump tuning range," Electron. Lett. 38, 85-86 (2002).
[CrossRef]

Kuhlmey, B. T.

Lamont, M. R. E.

Lin, C.

K. K. Chow, C. Shu, C. Lin, and A. Bjarklev, "Polarization-Insensitive Widely Tunable Wavelength Converter Based on Four-Wave Mixing in a Dispersion-Flattened Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 624-626 (2005).
[CrossRef]

T. Yang, C. Shu and C. Lin, "Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber," Optics Express 13, 5409-5415 (2005).
[CrossRef] [PubMed]

Lin, Q.

F. Yaman, Q. Lin, and G. P. Agrawal, "A novel design for polarization-independent single-pump fiber-optic parametric amplifier," IEEE Photonics Tech. Lett. 18, 2335-2337 (2006).
[CrossRef]

Luan, F.

Magi, E.

Mägi, E. C.

Marhic, M. E.

G. Kalogerakis and M. E. Marhic, "Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters," J. Lightwave Tech. 24, 3683-3691 (2006).
[CrossRef]

M. E. Marhic, F. S. Yang, Min-Chen Ho and L. G. kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Lightwave Tech. 17, 210-215 (1999).
[CrossRef]

M. E. Marhic, N. Kagi, T. K. Chiang, and L. G. Kazovsky, "Broadband fiber optical parametric amplifiers," Opt. Lett. 21, 573-575 (1996).
[CrossRef] [PubMed]

McKerracher, R. W.

McKinstrie, C. J.

M. Yu and C. J. McKinstrie, "Modulational instabilities in dispersion-flattened fibers," Physical Review E,  52, 1072-1080 (1995).
[CrossRef]

Moss, D. J.

Nagashima, T.

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

Nakazawa, M.

T. Yamamoto and M. Nakazawa, "Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching," IEEE Photonics Tech. Lett. 9, 327-329 (1997).
[CrossRef]

Ohara, S.

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

Pelusi, M. D.

Peucheret, C.

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
[CrossRef]

Roelens, M. A. F.

Sanghera, J. S.

Shaw, L. B.

Shu, C.

T. Yang, C. Shu and C. Lin, "Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber," Optics Express 13, 5409-5415 (2005).
[CrossRef] [PubMed]

K. K. Chow, C. Shu, C. Lin, and A. Bjarklev, "Polarization-Insensitive Widely Tunable Wavelength Converter Based on Four-Wave Mixing in a Dispersion-Flattened Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 624-626 (2005).
[CrossRef]

Sterke, C. M. D.

Sugimoto, N.

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

Tokle, T.

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
[CrossRef]

Westlund, M.

M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, "Transparent wavelength conversion in fibre with 24nm pump tuning range," Electron. Lett. 38, 85-86 (2002).
[CrossRef]

Yamamoto, T.

T. Yamamoto and M. Nakazawa, "Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching," IEEE Photonics Tech. Lett. 9, 327-329 (1997).
[CrossRef]

Yaman, F.

F. Yaman, Q. Lin, and G. P. Agrawal, "A novel design for polarization-independent single-pump fiber-optic parametric amplifier," IEEE Photonics Tech. Lett. 18, 2335-2337 (2006).
[CrossRef]

Yang, F. S.

M. E. Marhic, F. S. Yang, Min-Chen Ho and L. G. kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Lightwave Tech. 17, 210-215 (1999).
[CrossRef]

Yang, T.

T. Yang, C. Shu and C. Lin, "Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber," Optics Express 13, 5409-5415 (2005).
[CrossRef] [PubMed]

Yeom, Dong-II

Yu, M.

M. Yu and C. J. McKinstrie, "Modulational instabilities in dispersion-flattened fibers," Physical Review E,  52, 1072-1080 (1995).
[CrossRef]

Zhang, A.

Electron. Lett. (1)

M. Westlund, H. Hansryd, P. A. Andrekson, and S. N. Knudsen, "Transparent wavelength conversion in fibre with 24nm pump tuning range," Electron. Lett. 38, 85-86 (2002).
[CrossRef]

IEEE Photon. Tech. Lett. (1)

J. Hansryd and P. A. Andrekson, "Broad-Band Continuous-Wave-Pumped Fiber Optical Parametric Amplifier with 49-dB Gain and Wavelength-Conversion Efficiency," IEEE Photon. Tech. Lett. 13,194-196 (2001).
[CrossRef]

IEEE Photonics Tech. Lett. (4)

F. Yaman, Q. Lin, and G. P. Agrawal, "A novel design for polarization-independent single-pump fiber-optic parametric amplifier," IEEE Photonics Tech. Lett. 18, 2335-2337 (2006).
[CrossRef]

P. A. Andersen, T. Tokle, Y. Geng, C. Peucheret, and P. Jeppesen, "Wavelength Conversion of a 40-Gb/s RZ-DPSK Signal Using Four-Wave Mixing in a Dispersion-Flattened Highly Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 1908-1910 (2005).
[CrossRef]

K. K. Chow, C. Shu, C. Lin, and A. Bjarklev, "Polarization-Insensitive Widely Tunable Wavelength Converter Based on Four-Wave Mixing in a Dispersion-Flattened Nonlinear Photonic Crystal Fiber," IEEE Photonics Tech. Lett. 17, 624-626 (2005).
[CrossRef]

T. Yamamoto and M. Nakazawa, "Highly efficient four-wave mixing in an optical fiber with intensity dependent phase matching," IEEE Photonics Tech. Lett. 9, 327-329 (1997).
[CrossRef]

J. Lightwave Tech. (2)

G. Kalogerakis and M. E. Marhic, "Methods for full utilization of the bandwidth of fiber optical parametric amplifiers and wavelength converters," J. Lightwave Tech. 24, 3683-3691 (2006).
[CrossRef]

M. E. Marhic, F. S. Yang, Min-Chen Ho and L. G. kazovsky, "High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation," J. Lightwave Tech. 17, 210-215 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (5)

Opt. Lett. (4)

Optics Express (2)

T. Yang, C. Shu and C. Lin, "Depolarization technique for wavelength conversion using four-wave mixing in a dispersion-flattened photonic crystal fiber," Optics Express 13, 5409-5415 (2005).
[CrossRef] [PubMed]

K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, "Four-wave mixing based widely tunable wavelength conversion using 1-m dispersionshifted bismuth-oxide photonic crystal fiber," Optics Express 15, 15418-15423 (2007).
[CrossRef] [PubMed]

Physical Review E (1)

M. Yu and C. J. McKinstrie, "Modulational instabilities in dispersion-flattened fibers," Physical Review E,  52, 1072-1080 (1995).
[CrossRef]

Other (7)

M. Hirano, T. Nakanishi, T. Okuno, and Masashi Onishi, "Broadband wavelength conversion over 193-nm by HNL-DSF improving higher-order dispersion performance," in Proc. of ECOC 2005, Glasgow, Scotland, paper TH4.4.4 (2005)

J. Hiroishi, N. Kumano, K. Mukasa, R. Sugizaki, R. Miyabe, S. Matsushita, H. Tobioka, S. Namiki, and T. Yagi, "Dispersion slope controlled HNL-DSF with high γ of 25 W-1km-1 and band conversion experiment using this fiber," in Proc. of ECOC 2002, Copenhagen, Denmark, paper PD1.5 (2002).

J. H. Lee, T. Nagashima, T. Hasegawa, S. Ohara, N. Sugimoto, T. Tanemura, and K. Kikuchi, "Wavelength conversion of 40-Gbit/s NRZ signal using four-wave mixing in 40-cm-long bismuth oxide based highly-nonlinear optical fiber," in Proc. OFC 2005, paper PDP23, Anaheim, USA (2005).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001), Chap. 10.

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A. S. Lenihan and G. M. Carter, "Polarization-insensitive wavelength conversion at 40 Gb/s using birefringent nonlinear fiber," in Proc. Of CLEO 2007, Baltimore, USA, paper CThAA2 (2007)

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Figures (9)

Fig. 1.
Fig. 1.

The profile of Δβ(ωp ) with different ωs (the gray area represents exponential gain region)

Fig. 2.
Fig. 2.

The profile of Δβ(ωp ) when Φ < 0, Φ = 0 and Φ > 0 (ωs = ωopt s ).

Fig. 3.
Fig. 3.

Contours of the cutoff gain

Fig. 4.
Fig. 4.

Comparison between Ref. [2] and optimization results in different cases

Fig. 5.
Fig. 5.

(a): Measured contours of the cutoff gain Gc given in Ref. [9]. Converts from an arbitrary λs to any λi within the contours has exponential convert gain. (b): Theoretical contours of the cutoff gain. The side lengths of the black and red squares represent the maximal bandwidths of the one- and two-stage 1P-FPWC, respectively. Shadowed region represents the guard band against λ0 variations.

Fig. 6.
Fig. 6.

Schematic the WDM router based on 1P-FPWCs (FBG: fiber Bragg grating, TF: Tunable filter).

Fig. 7.
Fig. 7.

The bandwidth when β 4 = βopt 4 (a) and β 4 = 1.1βopt 4 (b). The shadowed region represents the guard band against λ 0 variations.

Fig. 8.
Fig. 8.

The variation of maximal bandwidth against the fourth order dispersion. The insets show the bandwidth when β 4 = 0.96βopt 4 and β 4 = 1.2βopt 4.

Fig. 9.
Fig. 9.

The profiles of Δβ(ωp ) with different ωs (the gray area represents exponential convert gain region)

Tables (3)

Tables Icon

Table 1. The corresponding ranges of ωs in Fig. 1

Tables Icon

Table 2. The parameters used in Fig. 4.

Tables Icon

Table 3. The corresponding ranges of a>s in Fig. 9

Equations (31)

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G = ( 1 + κ 2 4 g 2 ) sinh 2 ( gL ) ,
g = [ ( γ P 0 ) 2 ( κ / 2 ) 2 ] 1 / 2 .
κ = 2 γ P 0 + Δ β ,
Δ β = β s + β i 2 β p ,
Δ β = β 3 ( ω p ω 0 ) ( ω p ω s ) 2 + β 4 2 ( ω p ω 0 ) 2 ( ω p ω s ) 2 + β 4 12 ( ω p ω s ) 4
Δ β ( ω p ) = β 3 ( ω p ω 0 ) ( ω p ω s ) 2 .
ω e 1 = ω s ,
ω e 2 = ( 2 ω 0 + ω s ) / 3 .
Δ β ( ω e 1 ) = Δ β ( ω s ) = 0 ,
Δ β ( ω e 2 ) = 4 27 β 3 ( ω 0 ω s ) 3 .
ω p 1 = ω s and ω p 2 = ω 0 .
ω s = ω s opt = ω 0 3 ( γ P 0 / β 3 ) 1 / 3 .
ω p 3 = ω 0 4 ( γ P 0 / β 3 ) 1 / 3 , ω p 4 = ω 0 ( γ P 0 / β 3 ) 1 / 3 .
ω 0 4 ( γ P 0 / β 3 ) 1 / 3 ω p ω 0 .
Δ ω p max = 4 ( γ P 0 / β 3 ) 1 / 3 .
ω p 1 , p 2 = ω s ,
ω p 3 , p 4 = 1 7 β 4 ( 6 β 3 + 6 β 4 ω 0 + β 4 ω s ) ± 6 6 β 3 2 + 2 β 3 β 4 ( ω 0 ω s ) β 4 2 ( ω 0 ω s ) 2 .
ω 0 ( 1 + 7 ) β 3 / β 4 ω s ω 0 ( 1 7 ) β 3 / β 4 .
Δ ω p = ω p 3 ω p 4 = 2 6 7 β 4 6 β 3 2 + 2 β 3 β 4 ( ω 0 ω s ) β 4 2 ( ω 0 ω s ) 2 .
Δ ω p max = 2 6 7 β 3 β 4 ,
ω s = ω s opt = ω 0 β 3 / β 4 .
ω 0 ( 1 + 6 7 ) β 3 β 4 = ω p 4 ω p ω p 3 = ω 0 ( 1 6 7 ) β 3 β 4
ω p 5,6 = ω 0 β 3 β 4 ± 1 7 β 4 3 β 3 2 Φ ,
ω p 7,8 = ω 0 β 3 β 4 ± 1 7 β 4 3 β 3 2 + Φ ,
Φ = 9 β 3 4 336 P 0 β 4 3 γ < 0 .
β 4 = β 4 opt = ( 3 β 3 4 112 P 0 γ ) 1 / 3
Δ ω p , global max = 4 ( 96 / 7 ) 1 / 6 ( γ P 0 β 3 ) 1 / 3 .
ω s = ω p ± 6 Φ ' ± Φ ' 2 ( 4 / 3 ) β 4 γ P p β 4
ω e 1 = ω s ,
ω e 2 , e 3 = 1 14 β 4 ( 9 β 3 + 9 β 4 ω 0 + 5 β 4 ω s ± 3 27 β 3 2 + 2 β 3 β 4 ( ω 0 ω s ) β 4 2 ( ω 0 ω s ) 2 ) ,
ω 0 ( 1 + 2 7 ) β 3 / β 4 ω s ω 0 ( 1 2 7 ) β 3 / β 4 .

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