Abstract

A novel predictor-corrector method for the coupled equations for ultrabroad-band Raman amplifiers with multiple pumps is proposed and derived, for the first time, based on the Adams formula. The proposed algorithm is effective in solving Raman amplifier equations that include pumps, signals, noises, and their backscattering waves. The detail procedure is given, and proves the excellence of our algorithm. Simulation results show that, in designing the Raman amplifier, our multistep method can effectively improve the accuracy and stability compared with the one-step method and explicit multistep method. The numerical results show that the power of backscattering pumps and signals is lower by ~30 dB and 20 dB than their original power, respectively, and the power of forward and backward noises is less than that of input signals by ~30 dB under our simulation conditions.

© 2003 Optical Society of America

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2003 (9)

W. S. Wong, C. J. Chen, M. C. Ho, and H. K. Lee, “Phase-matched four-wave mixing between pumps and signals in a copumped Raman amplifier,” IEEE Photon. Technol. Lett. 15, 209–211 (2003).
[Crossref]

H. Kim and R. J. Essiambre, “Transmission of 8×20 Gb/s DQPSK signals over 310-km SMF with 0.8-b/s/Hz spectral efficiency,” IEEE Photon. Technol. Lett. 15, 769–771 (2003).
[Crossref]

P. C Xiao, Q. J zeng, J. Huang, and J. M. Liu, “A new optimal algorithm for multipump sources of distributed fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 206–208 (2003).
[Crossref]

N. Kikuchi, K. K. Wong, K. Uesaka, K. Shimizu, S. Yam, E. S. Hu, M. Marhic, and L. G. Kazovsky, “Novel in-service wavelength-band upgrade scheme for fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 27–29 (2003).
[Crossref]

B. Cuenot, “Comparison of engineering scenarios for N×160 Gb/s WDM transmission systems,” IEEE Photon. Technol. Lett. 15, 864–866 (2003).
[Crossref]

A. Pizzinat, M. Santagiustina, and C. Schivo, “Impact of hybrid EDFA-distributed Raman amplification on a 4×40-Gb/s WDM optical communication system,” IEEE Photon. Technol. Lett. 15, 341–343 (2003).
[Crossref]

X. M Liu, H. Y zhang, and Y. L Guo, “A novel method for Raman amplifier propagation equations,” IEEE Photon. Technol. Lett. 15, 392–394 (2003).
[Crossref]

X. M Liu and B Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11, 1452–1461 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1452
[Crossref] [PubMed]

C. Finot, G. Millot, C. Billet, and J. M. Dudley, “Experimental generation of parabolic pulses via Raman amplification in optical fiber,” Opt. Express 11, 1547–1552 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1547
[Crossref] [PubMed]

2002 (14)

V. E. Perlin and H. G. Winful, “Optimizing the noise performance of broad-band WDM systems with distributed Raman amplification,” IEEE Photon. Technol. Lett. 14, 1199–1201 (2002).
[Crossref]

S. Radic, S. Chandrasekhar, P. Bernasconi, J. Centanni, C. Abraham, N. Copner, and K. Tan, “Feasibility of hybrid Raman/EDFA amplification in bidirectional optical transmission,” IEEE Photon. Technol. Lett. 14, 221–223 (2002).
[Crossref]

T. E. Murphy, “10-GHz 1.3-ps pulse generation using chirped soliton compression in a Raman gain medium,” IEEE Photon. Technol. Lett. 14, 1424–1426 (2002).
[Crossref]

P. Kim, J. Park, H. Yoon, J. Park, and N. Park, “In situ design method for multichannel gain of a distributed Raman amplifier with multiwave OTDR,” IEEE Photon. Technol. Lett. 14, 1683–1685 (2002).
[Crossref]

M. E. Marhic and D. E. Nikonov, “Low third-order glass-host nonlinearities in erbium-doped waveguide amplifiers,” Proceedings of SPIE, vol.  4645, pp. 193 (2002).
[Crossref]

V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol. 20, 250–254 (2002).
[Crossref]

D. Dahan and G. Eisenstein, “Numerical comparison between distributed and discrete amplification in a point-to-point 40-gb/s 40-WDM-based transmission system with three different modulation formats,” J. Lightwave Technol. 20, 379–388 (2002).
[Crossref]

K. Suto, T. Saito, T. Kimura, J. I. Nishizawa, and T. Tanabe, “Semiconductor Raman amplifier for terahertz bandwidth optical communication,” J. Lightwave Technol. 20, 705–711 (2002).
[Crossref]

E. Poutrina and G. P. Agrawal, “Timing jitter in dispersion-managed soliton systems with distributed, lumped, and hybrid amplification,” J. Lightwave Technol. 20, 762–769 (2002).
[Crossref]

E. M. Dianov, “Advances in Raman fibers,” J. Lightwave Technol. 20, 1457–1462 (2002).
[Crossref]

M. Karásek and M. Menif, “Channel addition/removal response in Raman fiber amplifiers: modeling and experimentation,” J. Lightwave Technol. 20, 1680–1687 (2002).
[Crossref]

M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–559 (2002).
[Crossref]

K. Toge, K. Hogari, and T. Horiguchi, “Measurement of Raman gain distribution in optical fibers,” IEEE Photon. Technol. Lett. 14, 974–976 (2002).
[Crossref]

X. zhou, M. Birk, and S. Woodward, “Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifier,” IEEE Photon. Technol. Lett. 14,1686–1688 (2002).
[Crossref]

2001 (13)

B. Min, P. Kim, and N. Park, “Flat amplitude equal spacing 798-channel Rayleigh-assisted Brillouin/Raman multiwavelength comb generation in dispersion compensating fiber,” IEEE Photon. Technol. Lett. 13, 1352–1354 (2001).
[Crossref]

T. Okuno, T. Tsuzaki, and M. Nishimura, “Novel optical hybrid line configuration for quasi-lossless transmission by distributed Raman amplification,” IEEE Photon. Technol. Lett. 13, 806–808 (2001).
[Crossref]

L. D. Garrett, M. Eiselt, R. W. Tkach, V. Dominic, R. Waarts, D. Giltner, and D. Mehuys, “Field demonstration of distributed Raman amplification with 3.8-dB Q-improvement for 5×120-km transmission,” IEEE Photon. Technol. Lett. 13, 157–159 (2001).
[Crossref]

K. Song and S. D. Dods, “Cross modulation of pump-signals in distributed Raman amplifiers, theory and experiment,” IEEE Photon. Technol. Lett. 13, 1173–1175 (2001).
[Crossref]

A. Carena, V. Curri, and P. Poggiolini, “On the optimization of hybrid Raman/erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 13, 1170–1172 (2001).
[Crossref]

H. S. Seo, K. Oh, and U. C. Paek, “Gain optimization of germanosilicate fiber Raman amplifier and its applications in the compensation of Raman-induced crosstalk among wavelength division multiplexing channels,” IEEE J. Quantum Electron. 37, 1110–1116 (2001).
[Crossref]

L. Helczynski and A. Berntson, “Comparison of EDFA and bidirectionally pumped Raman amplifier in a 40-Gb/s Rz transmission system,” IEEE Photon. Technol. Lett. 13, 669–671 (2001).
[Crossref]

M. Achtenhagen, G. G. Change, B. Nyman, and A. Hardy, “Analysis of a multiple-pump Raman amplifier,” Appl. Phys. Lett,  78, 1322–1324 (2001).
[Crossref]

X. zhou, C. Lu, P. Shum, and T. H. Cheng, “A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,” IEEE Photon. Technol. Lett. 13, 945–947 (2001).
[Crossref]

Z. M. Liao and G. P. Agrawal, “Role of distributed amplification in designing high-capacity soliton systems.” Opt. Express 9, 66–71 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-2-66
[Crossref] [PubMed]

N. Takachio and H. Suzuki, “Application of Raman-distributed amplification to WDM transmission systems using 1.55-µm dispersion-shifted fiber,” J. Lightwave Technol. 19, 60–69 (2001).
[Crossref]

A. G. Okhrimchuk, G. Onishchukov, and E. Lederer, “Long-haul soliton transmission at 1.3 µm using distributed Raman amplification,” J. Lightwave Technol. 19, 837–841 (2001).
[Crossref]

C. M. McIntosh, A. G. Grandpierre, D. N. Christodoulides, J. Toulouse, and J. M. P. Delavaux, “Eliminating SRS channel depletion in massive WDM systems via optical filtering techniques,” IEEE Photon. Technol. Lett. 13, 302–304 (2001).
[Crossref]

2000 (1)

B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photon. Technol. Lett. 12, 1486–1488 (2000).
[Crossref]

1999 (1)

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett. 11, 530–532 (1999).
[Crossref]

1998 (1)

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

1997 (1)

P. B. Hansen, L. Eskildsen, S. G. Grubb, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Capacity upgrades of transmission systems by Raman amplification,” IEEE Photon. Technol. Lett. 9, 262–264 (1997).
[Crossref]

1996 (1)

D. N. Christodoulides and R. B. Jander, “Evolution of stimulated Raman crosstalk in wavelength division multiplexed systems,” IEEE Photon. Technol. Lett. 8, 1722–1724 (1996).
[Crossref]

1983 (1)

M. Nakazawa, “Rayleigh backscattering theory for single-mode optical fibers,” J. Opt. Soc. Amer. 73, 1175–1181 (1983).
[Crossref]

Abraham, C.

S. Radic, S. Chandrasekhar, P. Bernasconi, J. Centanni, C. Abraham, N. Copner, and K. Tan, “Feasibility of hybrid Raman/EDFA amplification in bidirectional optical transmission,” IEEE Photon. Technol. Lett. 14, 221–223 (2002).
[Crossref]

Achtenhagen, M.

M. Achtenhagen, G. G. Change, B. Nyman, and A. Hardy, “Analysis of a multiple-pump Raman amplifier,” Appl. Phys. Lett,  78, 1322–1324 (2001).
[Crossref]

Agrawal, G. P.

Bernasconi, P.

S. Radic, S. Chandrasekhar, P. Bernasconi, J. Centanni, C. Abraham, N. Copner, and K. Tan, “Feasibility of hybrid Raman/EDFA amplification in bidirectional optical transmission,” IEEE Photon. Technol. Lett. 14, 221–223 (2002).
[Crossref]

Berntson, A.

L. Helczynski and A. Berntson, “Comparison of EDFA and bidirectionally pumped Raman amplifier in a 40-Gb/s Rz transmission system,” IEEE Photon. Technol. Lett. 13, 669–671 (2001).
[Crossref]

Billet, C.

Birk, M.

X. zhou, M. Birk, and S. Woodward, “Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifier,” IEEE Photon. Technol. Lett. 14,1686–1688 (2002).
[Crossref]

Burden, R. L.

J. D. Faires and R. L. Burden, Numerical method. Boston (PWS-KENT Publishing Company, Boston, 1992), pp. 168–179.

Carena, A.

A. Carena, V. Curri, and P. Poggiolini, “On the optimization of hybrid Raman/erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 13, 1170–1172 (2001).
[Crossref]

Centanni, J.

S. Radic, S. Chandrasekhar, P. Bernasconi, J. Centanni, C. Abraham, N. Copner, and K. Tan, “Feasibility of hybrid Raman/EDFA amplification in bidirectional optical transmission,” IEEE Photon. Technol. Lett. 14, 221–223 (2002).
[Crossref]

Chandrasekhar, S.

S. Radic, S. Chandrasekhar, P. Bernasconi, J. Centanni, C. Abraham, N. Copner, and K. Tan, “Feasibility of hybrid Raman/EDFA amplification in bidirectional optical transmission,” IEEE Photon. Technol. Lett. 14, 221–223 (2002).
[Crossref]

Change, G. G.

M. Achtenhagen, G. G. Change, B. Nyman, and A. Hardy, “Analysis of a multiple-pump Raman amplifier,” Appl. Phys. Lett,  78, 1322–1324 (2001).
[Crossref]

Chen, C. J.

W. S. Wong, C. J. Chen, M. C. Ho, and H. K. Lee, “Phase-matched four-wave mixing between pumps and signals in a copumped Raman amplifier,” IEEE Photon. Technol. Lett. 15, 209–211 (2003).
[Crossref]

Cheng, T. H.

X. zhou, C. Lu, P. Shum, and T. H. Cheng, “A simplified model and optimal design of a multiwavelength backward-pumped Raman amplifier,” IEEE Photon. Technol. Lett. 13, 945–947 (2001).
[Crossref]

Christodoulides, D. N.

C. M. McIntosh, A. G. Grandpierre, D. N. Christodoulides, J. Toulouse, and J. M. P. Delavaux, “Eliminating SRS channel depletion in massive WDM systems via optical filtering techniques,” IEEE Photon. Technol. Lett. 13, 302–304 (2001).
[Crossref]

D. N. Christodoulides and R. B. Jander, “Evolution of stimulated Raman crosstalk in wavelength division multiplexed systems,” IEEE Photon. Technol. Lett. 8, 1722–1724 (1996).
[Crossref]

Copner, N.

S. Radic, S. Chandrasekhar, P. Bernasconi, J. Centanni, C. Abraham, N. Copner, and K. Tan, “Feasibility of hybrid Raman/EDFA amplification in bidirectional optical transmission,” IEEE Photon. Technol. Lett. 14, 221–223 (2002).
[Crossref]

Cuenot, B.

B. Cuenot, “Comparison of engineering scenarios for N×160 Gb/s WDM transmission systems,” IEEE Photon. Technol. Lett. 15, 864–866 (2003).
[Crossref]

Curri, V.

A. Carena, V. Curri, and P. Poggiolini, “On the optimization of hybrid Raman/erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 13, 1170–1172 (2001).
[Crossref]

Dahan, D.

Delavaux, J. M. P.

C. M. McIntosh, A. G. Grandpierre, D. N. Christodoulides, J. Toulouse, and J. M. P. Delavaux, “Eliminating SRS channel depletion in massive WDM systems via optical filtering techniques,” IEEE Photon. Technol. Lett. 13, 302–304 (2001).
[Crossref]

DeMarco, J. J.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

P. B. Hansen, L. Eskildsen, S. G. Grubb, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Capacity upgrades of transmission systems by Raman amplification,” IEEE Photon. Technol. Lett. 9, 262–264 (1997).
[Crossref]

Dianov, E. M.

DiGiovanni, D. J.

P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. 10, 159–161 (1998).
[Crossref]

P. B. Hansen, L. Eskildsen, S. G. Grubb, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Capacity upgrades of transmission systems by Raman amplification,” IEEE Photon. Technol. Lett. 9, 262–264 (1997).
[Crossref]

Dods, S. D.

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P. Kim, J. Park, H. Yoon, J. Park, and N. Park, “In situ design method for multichannel gain of a distributed Raman amplifier with multiwave OTDR,” IEEE Photon. Technol. Lett. 14, 1683–1685 (2002).
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P. B. Hansen, L. Eskildsen, S. G. Grubb, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Capacity upgrades of transmission systems by Raman amplification,” IEEE Photon. Technol. Lett. 9, 262–264 (1997).
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P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, “Rayleigh scattering limitations in distributed Raman pre-amplifiers,” IEEE Photon. Technol. Lett. 10, 159–161 (1998).
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Tanabe, T.

Tkach, R. W.

L. D. Garrett, M. Eiselt, R. W. Tkach, V. Dominic, R. Waarts, D. Giltner, and D. Mehuys, “Field demonstration of distributed Raman amplification with 3.8-dB Q-improvement for 5×120-km transmission,” IEEE Photon. Technol. Lett. 13, 157–159 (2001).
[Crossref]

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K. Toge, K. Hogari, and T. Horiguchi, “Measurement of Raman gain distribution in optical fibers,” IEEE Photon. Technol. Lett. 14, 974–976 (2002).
[Crossref]

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C. M. McIntosh, A. G. Grandpierre, D. N. Christodoulides, J. Toulouse, and J. M. P. Delavaux, “Eliminating SRS channel depletion in massive WDM systems via optical filtering techniques,” IEEE Photon. Technol. Lett. 13, 302–304 (2001).
[Crossref]

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T. Okuno, T. Tsuzaki, and M. Nishimura, “Novel optical hybrid line configuration for quasi-lossless transmission by distributed Raman amplification,” IEEE Photon. Technol. Lett. 13, 806–808 (2001).
[Crossref]

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L. D. Garrett, M. Eiselt, R. W. Tkach, V. Dominic, R. Waarts, D. Giltner, and D. Mehuys, “Field demonstration of distributed Raman amplification with 3.8-dB Q-improvement for 5×120-km transmission,” IEEE Photon. Technol. Lett. 13, 157–159 (2001).
[Crossref]

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V. E. Perlin and H. G. Winful, “Optimizing the noise performance of broad-band WDM systems with distributed Raman amplification,” IEEE Photon. Technol. Lett. 14, 1199–1201 (2002).
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V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol. 20, 250–254 (2002).
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N. Kikuchi, K. K. Wong, K. Uesaka, K. Shimizu, S. Yam, E. S. Hu, M. Marhic, and L. G. Kazovsky, “Novel in-service wavelength-band upgrade scheme for fiber Raman amplifier,” IEEE Photon. Technol. Lett. 15, 27–29 (2003).
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P. Kim, J. Park, H. Yoon, J. Park, and N. Park, “In situ design method for multichannel gain of a distributed Raman amplifier with multiwave OTDR,” IEEE Photon. Technol. Lett. 14, 1683–1685 (2002).
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Figures (7)

Fig. 1.
Fig. 1.

Illustration of frequency intervals for the numerical calculation. The entire spectral range of interest is represented by frequency v 0 to vn (Note that the horizontal axis in the ascending order of wavelength. Hence, the frequency is descending with horizontal axis, i.e., v 0 > vn .) (a) for input power P(0, vi ) at the position of z=0, and (b) for input power P(z, vi ) at the position of z=L.

Fig. 2.
Fig. 2.

Absolute error for one-step method, four-step method in [39] and PCM in this paper. The step size Δz=0.1.

Fig. 3.
Fig. 3.

The stability for one-step method, four-step method in [39] and PCM in this paper. The exact solution almost coincides with the solution of PCM.

Fig. 4.
Fig. 4.

Spectrum at the fiber input, output and backscattering, where (a) for input pump (backward) and signal (forward); (b) for output pump, backscattering signal and backward noise at the input-end of the fiber (i.e., z=0); (c) for output signal, forward noise and backscattering pump at the output-end of the fiber (i.e., z=L).

Fig. 5.
Fig. 5.

The propagation for signals, pumps, noises and their backscattering waves along the transmission fiber, where for (a) pumps, (b) backscattering pumps, (c) signals, (d) backscattering signals, (e) forward noises, and (f) backward noises. The fiber parts are 29. The dashes in Fig.5(a) are the projection of eight pumps.

Fig. 6.
Fig. 6.

Convergence for the reported four-step method [39] and our proposed PCM in designing DRA, where (a) for PCM and (b) for four-step method. All parameters are same with those in Fig.5 except that the single-pump power is 1.3 W at the wavelength of 1470 nm and there are only fifteen signals, from 192.15 THz to 194.95 THz.

Fig. 7.
Fig. 7.

Evolution of all signals along the fiber in each iteration under the starting boundary conditions of z=L (L=80 km). Figures (a), (b), (c), and (d) are the first-, second-, third-, and fourth-iteration for our algorithm, and their corresponding maximum of relative error of all signals are 0.3388, 0.0591, 0.0019, 9.844×10-4, respectively. The initially assumed value for each signal at z=80 km is 0.15 mW (i.e., -8.239 dBm), which are shown in figure (a). After only 4 iterations, the required relative error of <10-3 is contented [see figure (d)]. In the simulations, all parameters are same with Fig.5 and the noises are neglected (i.e., the first step of two-step procedure for the simulation algorithm in Section 3.2).

Equations (24)

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d p ± ( z , v i ) d z = α ( v i ) P ± ( z , v i ) ± η ( v i ) P ( z , v i )
± P ± ( z , v i ) m = 1 i 1 g R ( v m v i ) Γ A eff [ P ± ( z , v i ) + P ( z , v i ) ]
± h v i m = 1 i 1 g R ( v m v i ) Γ A eff [ P ± ( z , v i ) + P ( z , v i ) ] [ 1 + ( e h ( v m v i ) k T 1 ) 1 ] Δ v
P ± ( z , v i ) m = i + 1 n v i v m g R ( v i v m ) Γ A eff [ P ± ( z , v i ) + P ( z , v i ) ]
2 h v i P ± ( z , v i ) m = i + 1 n v i v m g R ( v i v m ) Γ A eff [ 1 + ( e h ( v i v m ) k T 1 ) 1 ] Δ μ
d P ± ( z , v i ) d z = P ± ( z , v i ) F ( z , v i ) ,
F ( z , v i ) = α ( v i ) ± η ( v i ) P ( z , v i ) P ± ( z , v i )
m = i + 1 n v i v m g R ( v i v m ) Γ A eff [ P ± ( z , v i ) + P ( z , v i ) + 2 h v i ( 1 + ( e h ( v i v m ) k T 1 ) 1 ) Δ μ ]
± m = 1 i 1 g R ( v m v i ) Γ A eff [ P ± ( z , v i ) + P ( z , v i ) ] [ 1 + h v i P ± ( z , v i ) ( 1 + ( e h ( v m v i ) k T 1 ) 1 ) Δ v ]
P ± ( z j + 1 , v ) = P ± ( z j , v ) exp ( F ( z j , v ) Δ z ) ,
P ± ( z j + 1 , v ) ¯ = P ± ( z j , v ) exp [ ( 55 · F ( z j , v ) 59 · F ( z j 1 , v ) + 37 · F ( z j 2 , v ) 9 · F ( z j 3 , v ) ) Δ z 24 ] .
P ± ( z j + 1 , v ) = P ± ( z j , v ) exp [ ( 9 · F ( z j + 1 , v ) ¯ + 19 · F ( z j , v ) 5 · F ( z j 1 , v ) + F ( z j 2 , v ) ) Δ z 24 ] ,
P ± ( z 1 , v ) = P ± ( z 0 , v ) exp ( F ( z 0 , v ) Δ z ) ,
P ± ( z 2 , v ) ¯ = P ± ( z 1 , v ) exp [ ( 3 · F ( z 1 , v ) F ( z 0 , v ) ) Δ z 2 ] ,
P ± ( z 2 , v ) = P ± ( z 1 , v ) exp ( F ( z 2 , v ) ¯ + F ( z 1 , v ) ) Δ z 2 ,
P ± ( z 3 , v ) ¯ = P ± ( z 2 , v ) exp [ ( 23 · F ( z 2 , v ) 16 · F ( z 1 , v ) + 5 F ( z 0 , v ) ) Δ z 12 ] ,
P ± ( z 3 , v ) = P ± ( z 2 , v ) exp [ ( 5 · F ( z 3 , v ) ¯ + 8 · F ( z 2 , v ) F ( z 1 , v ) ) Δ z ) 12 ] ,
d y d z = y + z + 1 , and y ( 0 ) = 1 .
d y d z = y · ln ( y ) · ( z 2 + z + 1 ) , and y ( 0 ) = 2 .
y ( z ) = exp { exp [ 1 3 z 3 + 1 2 z 2 + z + ln ( ln ( 2 ) ) ] } .
P ± ( z j + 1 , v ) ¯ = P ± ( z j , v ) exp [ ( 1901 · F ( z j , v ) 2774 · F ( z j 1 , v )
+ 2616 · F ( z j 2 , v ) 1274 · F ( z j 3 , v ) + 251 · F ( z j 4 , v ) ) Δ z 720 ] ,
P ± ( z j + 1 , v ) = P ± ( z j , v ) exp [ ( 251 · F ( z j + 1 , v ) ¯ + 646 · F ( z j , v )
264 · F ( z j 1 , v ) + 106 · F ( z j 2 , v ) 19 · F ( z j 3 , v ) ) Δ z 720 ] .

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