Abstract

An ideal optical backpropagation (OBP) scheme to compensate for dispersion and nonlinear effects of the transmission fibers is proposed. The scheme consists of an optical phase conjugator (OPC), N spans of dispersion-decreasing fibers (DDFs) and amplifiers, placed at the end of the fiber optic link. In order to compensate for the nonlinear effects of the transmission fibers exactly, the nonlinear coefficient of the backpropagation fiber has to increase exponentially with distance or equivalently the power in the backpropagation fiber should increase exponentially with distance if the nonlinear coefficient is constant. In this paper, it is shown that a combination of DDFs and amplifiers can compensate for the nonlinear effects exactly. An analytical expression for the dispersion profile of the DDF is derived. Numerical simulation of a long haul wavelength division multiplexing (WDM) fiber optic system with the proposed OBP scheme shows that the system reach can be enhanced by 54% as compared to digital backpropagation (DBP).

© 2013 Optical Society of America

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    [CrossRef]

2013

M. D. Pelusi, “WDM signal all-optical pre-compensation of the fiber nonlinearity in dispersion-managed links,” IEEE Photonics Technol. Lett.25(1), 71–74 (2013).
[CrossRef]

S. Kumar and J. Shao, “Optical back propagation with optimal step size for fiber optic transmission systems,” IEEE Photonics Technol. Lett.25(5), 523–526 (2013).
[CrossRef]

2012

2011

2010

2009

T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.27(8), 989–999 (2009).
[CrossRef]

C. Xie, “Inter-channel nonlinearities in coherent polarization-division-multiplexed quadrature-phase-shift-keying systems,” IEEE Photonics Technol. Lett.21(5), 274–276 (2009).
[CrossRef]

2008

2006

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

2005

2001

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

1996

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol.14(3), 243–248 (1996).
[CrossRef]

1995

1993

1990

1979

Asif, R.

Bayvel, P.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Behrens, C.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Boyd, R. W.

Cartledge, J. C.

Chen, X.

Chernikov, S. V.

Chowdhury, D.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Conradi, J.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Cristiani, I.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

Culverhouse, D.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Degiorgio, V.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

Dianov, E. M.

Du, L. B.

Ellis, A. D.

Ennser, K.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Evans, A. F.

Fejer, M. M.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

Fekete, D.

Gao, G.

Gao, Y.

Giroux, C.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Goldfarb, G.

Gordon, J. P.

Hallock, B.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Hellerbrand, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Hoffmann, S.

Holtmannspoetter, M.

Ip, E.

Kahn, J. M.

Ke, J. H.

Kennedy, T.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Killey, R. I.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Kim, I.

Kruse, A.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Kumar, S.

S. Kumar and J. Shao, “Optical back propagation with optimal step size for fiber optic transmission systems,” IEEE Photonics Technol. Lett.25(5), 523–526 (2013).
[CrossRef]

J. Shao and S. Kumar, “Optical backpropagation for fiber-optic communications using optical phase conjugation at the receiver,” Opt. Lett.37(15), 3012–3014 (2012).
[CrossRef] [PubMed]

S. Kumar and D. Yang, “Optical backpropagation for fiber-optic communications using highly nonlinear fibers,” Opt. Lett.36(7), 1038–1040 (2011).
[CrossRef] [PubMed]

S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett.30(24), 3278–3280 (2005).
[CrossRef] [PubMed]

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Langrock, C.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

Lascar, N.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Li, G.

Li, X.

Lin, C. Y.

Lowery, A. J.

Makovejs, S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Marazzi, L.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

Martinelli, M.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

Mateo, E.

Mateo, E. F.

Millar, D. S.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Minzioni, P.

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

Mollenauer, L. F.

Noé, R.

Payne, D. N.

Pelusi, M. D.

M. D. Pelusi, “WDM signal all-optical pre-compensation of the fiber nonlinearity in dispersion-managed links,” IEEE Photonics Technol. Lett.25(1), 71–74 (2013).
[CrossRef]

Pepper, D. M.

Pfau, T.

Rafique, D.

Richardson, D. J.

Roudas, I.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Savory, S. J.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

Schmauss, B.

Shao, J.

S. Kumar and J. Shao, “Optical back propagation with optimal step size for fiber optic transmission systems,” IEEE Photonics Technol. Lett.25(5), 523–526 (2013).
[CrossRef]

J. Shao and S. Kumar, “Optical backpropagation for fiber-optic communications using optical phase conjugation at the receiver,” Opt. Lett.37(15), 3012–3014 (2012).
[CrossRef] [PubMed]

Sharma, M.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Shieh, W.

Shirasaki, M.

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol.14(3), 243–248 (1996).
[CrossRef]

Stentz, A. J.

Tomkos, I.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Vodhanel, R. S.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Wang, C.-C.

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

Watanabe, S.

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol.14(3), 243–248 (1996).
[CrossRef]

Xie, C.

C. Xie, “Inter-channel nonlinearities in coherent polarization-division-multiplexed quadrature-phase-shift-keying systems,” IEEE Photonics Technol. Lett.21(5), 274–276 (2009).
[CrossRef]

Yam, S. S.-H.

Yaman, F.

Yang, D.

Yariv, A.

Zhong, K. P.

Zhu, L.

IEEE J. Sel. Top. Quantum Electron.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron.16(5), 1217–1226 (2010).
[CrossRef]

I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron.7(3), 439–460 (2001).
[CrossRef]

IEEE Photonics Technol. Lett.

C. Xie, “Inter-channel nonlinearities in coherent polarization-division-multiplexed quadrature-phase-shift-keying systems,” IEEE Photonics Technol. Lett.21(5), 274–276 (2009).
[CrossRef]

P. Minzioni, I. Cristiani, V. Degiorgio, L. Marazzi, M. Martinelli, C. Langrock, and M. M. Fejer, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photonics Technol. Lett.18(9), 995–997 (2006).
[CrossRef]

M. D. Pelusi, “WDM signal all-optical pre-compensation of the fiber nonlinearity in dispersion-managed links,” IEEE Photonics Technol. Lett.25(1), 71–74 (2013).
[CrossRef]

S. Kumar and J. Shao, “Optical back propagation with optimal step size for fiber optic transmission systems,” IEEE Photonics Technol. Lett.25(5), 523–526 (2013).
[CrossRef]

J. Lightwave Technol.

Opt. Express

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express18(14), 15144–15154 (2010).
[CrossRef] [PubMed]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express16(2), 880–888 (2008).
[CrossRef] [PubMed]

E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express16(20), 16124–16137 (2008).
[CrossRef] [PubMed]

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express18(16), 17075–17088 (2010).
[CrossRef] [PubMed]

R. Asif, C. Y. Lin, M. Holtmannspoetter, and B. Schmauss, “Optimized digital backward propagation for phase modulated signals in mixed-optical fiber transmission link,” Opt. Express18(22), 22796–22807 (2010).
[CrossRef] [PubMed]

D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express19(4), 3449–3454 (2011).
[CrossRef] [PubMed]

G. Gao, X. Chen, and W. Shieh, “Influence of PMD on fiber nonlinearity compensation using digital back propagation,” Opt. Express20(13), 14406–14418 (2012).
[CrossRef] [PubMed]

Opt. Lett.

Other

J. Conradi, S. Kumar, and S. S. Rosenblum, “Negative dispersion single mode waveguide fiber,” US patent 6430346 B1, (2002).

M. Morshed, L. B. Du, B. Foo, M. D. Pelusi, and A. J. Lowery, “Optical phase conjugation for nonlinearity compensation of 1.21-Tb/s Pol-Mux coherent optical OFDM,” in 18th OptoElectronics and Communications Conference (2013), paper PD3–4.

D. Rafique and A. D. Ellis, “Nonlinearity compensation via spectral inversion and digital back-propagation: A practical approach,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OM3A.1.
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Figures (5)

Fig. 1
Fig. 1

A single-span fiber optic system with (a) OBP using an ideal optical backpropagation fiber with negative loss coefficient; (b) OBP using a DDF and amplifiers. Tx: transmitter; TF: transmission fiber; OPC: optical phase conjugator; OBPF: optical backpropagation fiber; DDF: dispersion-decreasing fiber; Rx: receiver.

Fig. 2
Fig. 2

Dispersion profiles of DDF. TF parameters: α = 0.2 dB/km, β2 = 5 ps2/km, γ = 2.2 W−1km−1, La = 60 km. DDF parameters: αd = 0.4 dB/km, γd = 4.86 W−1km−1. (a) G = 1.0: β2,d(0) = 175.1 ps2/km, Ld = 20.5 km; (b) G = 1.26: β2,d(0) = 220.6 ps2/km, Ld = 12.1 km; (c) G = 1.5: β2,d(0) = 262.6 ps2/km, Ld = 9.0 km.

Fig. 3
Fig. 3

Schematic diagram of a WDM fiber optic transmission system with OBP. MUX: multiplexer; BPF: band pass filter; DMUX: demultiplexer.

Fig. 4
Fig. 4

BER versus launch power per WDM channel. Transmission distance = 1200 km.

Fig. 5
Fig. 5

BERmin versus transmission distance.

Equations (25)

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q z =i[ D(t)+N(t,z) ]q(t,z),
D(t)= β 2 2 2 t 2 ,N(t,z)=γ | q(t,z) | 2 +i α 2 ,
q(t, L a )=exp{ i 0 L a [ D(t)+N(t,z) ]dz }q(t,0).
q (t, L a )=exp{ i 0 L a [ D(t)+ N (t,z) ]dz } q (t,0).
q OBPF,out (t)= e i 0 L a [ D(t)+ N (t,z) ]dz q (t, L a )
= e i 0 L a [ D(t)+ N (t,z) ]dz e i 0 L a [ D(t)+ N (t,z) ]dz q (t,0) = q * (t,0).  
q b z b =i[ D(t)+ N (t, z b ) ] q b (t, z b ),
q b = P in e α( L a z b )/2 u b ,
d z b = β 2 d z b ,
i u b z b 1 2 2 u b t 2 + γ P in β 2 e α( L a z b ) | u b | 2 u b =0,
i q b z d β 2,d ( z d ) 2 2 q b t 2 + γ d | q b | 2 q b +i α d 2 q b =0,
q b = P d e α d z d /2 u b ,
d z d = β 2,d ( z d )d z d ,
i u b z d 1 2 2 u b t 2 + γ d P d e α d z d β 2,d ( z d ) | u b | 2 u b =0,
d z b =d z d ,
γ P in β 2 e α( L a z b ) = γ d P d e α d z d β 2,d ( z d ) .
β 2 d z b d z d = β 2,d ( z d ),
w β 2 z b = 0 z d β 2,d ( z d ) d z d ,
dw d z d = β 2,d ( z d ).
dw d z d e αw/ β 2 =( γ d P d β 2 γ P in ) e α L a e α d z d .
β 2 α ( e α β 2 w( z d ) 1 )=( γ d P d γ P in ) e α L a 1 e α d z d α d .
w( z d )= β 2 α ln{ 1+ γ d Gα γ e α L a 1 e α d z d α d },
β 2,d ( z d )= e α d z d γ e α L a γ d G +α( 1 e α d z d α d ) β 2 .
β 2 L a =w( L d )= 0 L d β 2,d ( z d )d z d ,
L d = 1 α d ln{ 1 α d γ e α L a γ d Gα ( e α L a 1 ) }.

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