Abstract

An optical back propagation (OBP) technique is investigated to compensate for nonlinear impairments in fiber optic communication systems with reconfigurable optical add-drop multiplexers (ROADMs). An OBP module consisting of an optical phase conjugator (OPC), amplifiers and dispersion-decreasing fibers (DDFs) fully compensates for the nonlinear impairments of a transmission fiber. The OBP module can be placed after each transmission fiber (inline OBP case) or at each network node (node OBP case). For a wavelength division multiplexing (WDM) system with 2400 km transmission distance and 32-quadrature amplitude modulation (QAM) format, inline OBP and node OBP bring Q-factor improvements of 4.9 dB and 5.6 dB as compared with linear compensation, respectively. In contrast, receiver-side digital back propagation (DBP) only provides 1.3 dB Q-factor gain, due to its incapability of mitigating inter-channel nonlinear effects in fiber optic networks.

© 2016 Optical Society of America

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References

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2015 (1)

2014 (2)

2013 (3)

X. Liang, S. Kumar, and J. Shao, “Ideal optical backpropagation of scalar NLSE using dispersion-decreasing fibers for WDM transmission,” Opt. Express 21(23), 28668–28675 (2013).
[Crossref]

M. D. Pelusi, “WDM signal all-optical pre-compensation of the fiber nonlinearity in dispersion-managed links,” IEEE Photon. 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 Photon. Technol. Lett. 25(5), 523–526 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (1)

2009 (1)

2008 (3)

2006 (1)

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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

2002 (1)

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quant. Electron.,  26(3), 131–191 (2002).
[Crossref]

1996 (2)

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]

A. Hasegawa, S. Kumar, and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett. 21(1), 39–41 (1996).
[Crossref] [PubMed]

1995 (1)

1994 (1)

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Experimental demonstration of step-like dispersion profiling in optical fibre for soliton pulse generation and compression,” Electron. Lett. 30(5), 433–435 (1994).
[Crossref]

1993 (1)

1990 (1)

1979 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Borowiec, A.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. OâĂŹSullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Boyd, R. W.

Cartledge, J. C.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. OâĂŹSullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Chen, X.

Chernikov, S. V.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Experimental demonstration of step-like dispersion profiling in optical fibre for soliton pulse generation and compression,” Electron. Lett. 30(5), 433–435 (1994).
[Crossref]

S. V. Chernikov, D. J. Richardson, D. N. Payne, and E. M. Dianov, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18(7), 476–478 (1993).
[Crossref] [PubMed]

Cong, G.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

Deen, M. J.

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley, 2014), Chap. 10 and 11.
[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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

Dianov, E. M.

Dou, L.

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

Du, L. B.

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

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 Proceedings of 18th OptoElectronics and Communications Conference, (2013), paper PD3-4.

Evans, A. F.

Fan, Y.

Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

Fekete, D.

Foo, B.

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 Proceedings of 18th OptoElectronics and Communications Conference, (2013), paper PD3-4.

Gao, Y.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. OâĂŹSullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Goldfarb, G.

Gordon, J. P.

Hasegawa, A.

He, G. S.

G. S. He, “Optical phase conjugation: principles, techniques, and applications,” Prog. Quant. Electron.,  26(3), 131–191 (2002).
[Crossref]

Hoffmann, S.

Hoshida, T.

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

Ikeda, K.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

Ip, E.

Kahn, J. M.

Karar, A. S.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. OâĂŹSullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Kashyap, R.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Experimental demonstration of step-like dispersion profiling in optical fibre for soliton pulse generation and compression,” Electron. Lett. 30(5), 433–435 (1994).
[Crossref]

Kawashima, H.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

Kim, I.

Kodama, Y.

Kumar, S.

X. Liang and S. Kumar, “Correlated digital back propagation based on perturbation theory,” Opt. Express 23(11), 14655–14665 (2015).
[Crossref] [PubMed]

X. Liang, S. Kumar, J. Shao, M. Malekiha, and D. V. Plant, “Digital compensation of cross-phase modulation distortions using perturbation technique for dispersion-managed fiber-optic systems,” Opt. Express 22(17), 20634–20645 (2014).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Multi-stage perturbation theory for compensating intra-channel nonlinear impairments in fiber-optic links,” Opt. Express 22(24), 29733–29745 (2014).
[Crossref]

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

X. Liang, S. Kumar, and J. Shao, “Ideal optical backpropagation of scalar NLSE using dispersion-decreasing fibers for WDM transmission,” Opt. Express 21(23), 28668–28675 (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]

A. Hasegawa, S. Kumar, and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett. 21(1), 39–41 (1996).
[Crossref] [PubMed]

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley, 2014), Chap. 10 and 11.
[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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

Laperle, C.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. OâĂŹSullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Li, G.

Li, L.

Li, X.

Liang, X.

Lowery, A. J.

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

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 Proceedings of 18th OptoElectronics and Communications Conference, (2013), paper PD3-4.

Malekiha, M.

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 Photon. 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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

Mateo, E.

Matsuura, H.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

Mollenauer, L. F.

Morshed, M.

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 Proceedings of 18th OptoElectronics and Communications Conference, (2013), paper PD3-4.

Nakashima, H.

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Namiki, S.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

Noé, R.

OâAZSullivan, M.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. OâĂŹSullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Oda, S.

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Oyama, T.

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Payne, D. N.

Pelusi, M. D.

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

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 Proceedings of 18th OptoElectronics and Communications Conference, (2013), paper PD3-4.

Pepper, D.M.

Pfau, T.

Plant, D. V.

Rasmussen, J. C.

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

Richardson, D. J.

Roberts, K.

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. OâĂŹSullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Shao, J.

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.

Suda, S.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

Suzuki, K.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

Tanizawa, K.

H. Kawashima, K. Suzuki, K. Tanizawa, S. Suda, G. Cong, H. Matsuura, S. Namiki, and K. Ikeda, “Multi-port optical switch based on silicon photonics,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2016), paper W1E.6.

Tao, Z.

Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. 29(17), 2570–2576 (2011).
[Crossref]

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

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T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

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Electron. Lett. (1)

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Y. Fan, L. Dou, Z. Tao, T. Hoshida, and J. C. Rasmussen, “A high performance nonlinear compensation algorithm with reduced complexity based on XPM model,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

T. Oyama, H. Nakashima, S. Oda, T. Yamauchi, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Robust and efficient receiver-side compensation method for intra-channel nonlinear effects,” in Proceedings of Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

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Figures (7)

Fig. 1
Fig. 1 Schematic of a fiber optic network with ROADMs.
Fig. 2
Fig. 2 Schematics of back propagation based on (a) a virtual fiber and (b) a DDF. Tx: transmitter, Rx: receiver, TF: transmission fiber, OPC: optical phase conjugator, DDF: dispersion-decreasing fiber.
Fig. 3
Fig. 3 (a) OPC based on a HNLF, (b) OPC with one laser pump, (c) OPC with two laser pumps, (d) OBP module for multi-channel systems. MUX: multiplexer, HNLF: highly nonlinear fiber, BPF: band pass filter, DeMUX: demultiplexer, OPC: optical phase conjugator, DDF: dispersion-decreasing fiber. ω0 is the carrier frequency of the optical signal.
Fig. 4
Fig. 4 Fiber-optic mesh networks using OBP to compensate for signal propagation impairments. (a) Inline OBP, (b) Node OBP. Tx: transmitter, Rx: receiver, TF: transmission fiber, OBP: optical back propagation, ROADM: reconfigurable optical add-drop multiplexer.
Fig. 5
Fig. 5 DDF dispersion profile fluctuations. The dispersion profile is modeled as: β ^ 2 , d ( z ) = [ 1 + x ( z ) ] β 2 , d ( z ), where β2,d (z) is a desired dispersion profile (ideal case), x(z) are zero-mean Gaussian random variables with a standard deviation of σDDF.
Fig. 6
Fig. 6 Q-factor vs. DDF dispersion profile fluctuation. Inline OBP using one laser pump, transmission distance = 2400 km, launch power per WDM channel = −2 dBm.
Fig. 7
Fig. 7 Q-factor vs. launch power per WDM channel. transmission distance = 2400 km, σDDF = 0.6 for non-ideal DDF cases.

Equations (23)

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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 ) ] d z } q ( t , 0 ) ,
q * ( t , L a ) = exp { i 0 L a [ D ( t ) + N * ( t , z ) ] d z } q * ( t , 0 ) .
D v ( t ) = β 2 2 t , N v ( t ) = γ | q ( t , z ) | 2 i α 2 ,
q out ( t ) = exp { i 0 L a [ D v ( t ) + N v ( t , z ) ] d z } q * ( t , L a ) .
q out ( t ) = q * ( t , 0 ) .
β 2 , d ( z ) = e α d z γ e α L a γ d G + α ( 1 e α d z α d ) β 2 ,
L d = 1 α d l n [ 1 α d γ e α L a γ d G α ( e α L a 1 ) ] .
q = q s + q p e i Ω p t ,
i γ | q | 2 q = i γ { | q s | 2 q s + 2 | q p | 2 q s + q s 2 q p * e i Ω p t + 2 | q s | 2 q p e i Ω p t + | q p | 2 q p e i Ω p t + q p 2 q s * e i 2 Ω p t } ,
q z + α 2 q = i γ | q | 2 q .
q c z + α 2 q c = i γ q p 2 q s * .
q p = A p e α z / 2 , A p = P p ,
q s ( t , z ) = A s ( t , z ) e α z / 2 .
q c ( t , L ) = i γ P p A s * L e f f e α L / 2 ,
P out | q c ( t , L ) | 2 = ( γ P p L e f f ) 2 e α L P in ,
loss 1 = P out P in = ( γ P p L e f f ) 2 e α L .
q = q s + q p 1 e i Ω 1 t + q p 2 e i Ω 2 t .
q c ( t , L ) = 2 i γ A p 1 A p 2 A s * L e f f e α L / 2 ,
loss 2 = 4 P p 1 P p 2 ( γ L e f f ) 2 e α L .
A p = ( A p 0 + δ A p 0 ) e i θ p ,
β ^ 2 , d ( z ) = [ 1 + x ( z ) ] β 2 , d ( z ) ,

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