Abstract

We have investigated an optical back propagation (OBP) method to compensate for propagation impairments in fiber optic networks with lumped Erbium doped fiber amplifier (EDFA) and/or distributed Raman amplification. An OBP module consists of an optical phase conjugator (OPC), optical amplifiers and dispersion varying fibers (DVFs). We derived a semi-analytical expression that calculates the dispersion profile of DVF. The OBP module acts as a nonlinear filter that fully compensates for the nonlinear distortions due to signal propagation in a transmission fiber, and is applicable for fiber optic networks with reconfigurable optical add-drop multiplexers (ROADMs). We studied a wavelength division multiplexing (WDM) network with 3000 km transmission distance and 64-quadrature amplitude modulation (QAM) modulation. OBP brings 5.8 dB, 5.9 dB and 6.1 dB Q-factor gains over linear compensation for systems with full EDFA amplification, hybrid EDFA/Raman amplification, and full Raman amplification, respectively. In contrast, digital back propagation (DBP) or OPC-only systems provide only 0.8 ~ 1.5 dB Q-factor gains.

© 2017 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  2. R.J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” IEEE Electron. Lett. 35(18), 1576–1578 (1999).
    [Crossref]
  3. A. Mecozzi, C.B. Clausen, and M. Shtaif, “System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(12), 1633–1635 (2000).
    [Crossref]
  4. S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett. 13(8), 800–802 (2001).
    [Crossref]
  5. P. Poggiolini, “The GN model of nonlinear propagation in uncompensated coherent optical systems,” J. Lightwave Technol. 30(24), 3857–3879 (2012).
    [Crossref]
  6. S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley, 2014).
    [Crossref]
  7. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990).
    [Crossref] [PubMed]
  8. K. P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. 22(3), 779–783 (2004).
    [Crossref]
  9. S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005).
    [Crossref]
  10. R. J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in European Conference on Optical Communication (ECOC 2005), paper Tu 3.2.2.
  11. K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
    [Crossref]
  12. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [Crossref]
  13. 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. Express 16(2), 880–888 (2008).
    [Crossref] [PubMed]
  14. 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. Express 16(20), 16124–16137 (2008).
    [Crossref] [PubMed]
  15. 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]
  16. 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]
  17. J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
    [Crossref]
  18. 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.
  19. 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.
  20. Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.
  21. 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]
  22. 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]
  23. X. Liang and S. Kumar, “Correlated digital back propagation based on perturbation theory,” Opt. Express 23(11), 14655–14665 (2015).
    [Crossref] [PubMed]
  24. A. Yariv, D. Fekete, and D.M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979).
    [Crossref] [PubMed]
  25. 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]
  26. 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]
  27. K. Solis-Trapala, T. Inoue, and S. Namiki, “Nearly-ideal optical phase conjugation based nonlinear compensation system,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper W3F.8.
  28. S. Kumar and D. Yang, “Optical backpropagation for fiber-optic communications using highly nonlinear fibers,” Opt. Lett. 36(7), 1038–1040 (2011).
    [Crossref] [PubMed]
  29. 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]
  30. 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.
  31. 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]
  32. 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]
  33. 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]
  34. B. Foo, B. Corcoran, C. Zhu, and A. J. Lowery, “Distributed nonlinear compensation of dual-polarization signals using optoelectronics,” IEEE Photon. Technol. Lett. 28(20), 2141–2144 (2016).
    [Crossref]
  35. X. Liang and S. Kumar, “Optical back propagation for compensating nonlinear impairments in fiber optic links with ROADMs,” Opt. Express 24(20), 22682–22692 (2016).
    [Crossref] [PubMed]
  36. S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).
  37. M. Morshed, L. B. Du, and A. J. Lowery, “Mid-span spectral inversion for coherent optical OFDM systems: fundamental limits to performance,” J. Lightwave Technol. 31(1), 58–66 (2013).
    [Crossref]
  38. 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]
  39. C. R. Menyuk, “Interaction of nonlinearity and polarization mode dispersion,” J. Opt. Fiber Commun. Rep. 1(4), 305–311 (2004).
    [Crossref]
  40. S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
    [Crossref] [PubMed]
  41. A. D. Ellis and M. McCarthy, “Impact of optical phase conjugation on the Shannon capacity limit,” in Optical Fiber Communication Conference, (Optical Society of America, 2016), paper Th4F.2.

2016 (2)

B. Foo, B. Corcoran, C. Zhu, and A. J. Lowery, “Distributed nonlinear compensation of dual-polarization signals using optoelectronics,” IEEE Photon. Technol. Lett. 28(20), 2141–2144 (2016).
[Crossref]

X. Liang and S. Kumar, “Optical back propagation for compensating nonlinear impairments in fiber optic links with ROADMs,” Opt. Express 24(20), 22682–22692 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (2)

2013 (5)

M. Morshed, L. B. Du, and A. J. Lowery, “Mid-span spectral inversion for coherent optical OFDM systems: fundamental limits to performance,” J. Lightwave Technol. 31(1), 58–66 (2013).
[Crossref]

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (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]

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]

2012 (2)

2011 (2)

2010 (1)

2009 (1)

2008 (3)

2007 (1)

2006 (2)

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]

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[Crossref]

2005 (1)

2004 (2)

K. P. Ho and J. M. Kahn, “Electronic compensation technique to mitigate nonlinear phase noise,” J. Lightwave Technol. 22(3), 779–783 (2004).
[Crossref]

C. R. Menyuk, “Interaction of nonlinearity and polarization mode dispersion,” J. Opt. Fiber Commun. Rep. 1(4), 305–311 (2004).
[Crossref]

2001 (1)

S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett. 13(8), 800–802 (2001).
[Crossref]

2000 (1)

A. Mecozzi, C.B. Clausen, and M. Shtaif, “System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(12), 1633–1635 (2000).
[Crossref]

1999 (1)

R.J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” IEEE Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

1996 (1)

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]

1990 (1)

1979 (1)

Agrawal, G. P.

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

Bickham, S. R.

S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).

Borowiec, A.

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

Cain, M. B.

S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).

Cartledge, J. C.

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

Chen, X.

Clausen, C.B.

A. Mecozzi, C.B. Clausen, and M. Shtaif, “System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(12), 1633–1635 (2000).
[Crossref]

Corcoran, B.

B. Foo, B. Corcoran, C. Zhu, and A. J. Lowery, “Distributed nonlinear compensation of dual-polarization signals using optoelectronics,” IEEE Photon. Technol. Lett. 28(20), 2141–2144 (2016).
[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 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).
[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]

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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

Du, L. B.

Ellis, A. D.

A. D. Ellis and M. McCarthy, “Impact of optical phase conjugation on the Shannon capacity limit,” in Optical Fiber Communication Conference, (Optical Society of America, 2016), paper Th4F.2.

Essiambre, R. J.

R. J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in European Conference on Optical Communication (ECOC 2005), paper Tu 3.2.2.

Essiambre, R.J.

R.J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” IEEE Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

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 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.

B. Foo, B. Corcoran, C. Zhu, and A. J. Lowery, “Distributed nonlinear compensation of dual-polarization signals using optoelectronics,” IEEE Photon. Technol. Lett. 28(20), 2141–2144 (2016).
[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 18th OptoElectronics and Communications Conference, (2013), paper PD3-4.

Gao, Y.

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

Goldfarb, G.

Gordon, J. P.

Hardcastle, I.

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[Crossref]

Ho, K. P.

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]

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 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Inoue, T.

K. Solis-Trapala, T. Inoue, and S. Namiki, “Nearly-ideal optical phase conjugation based nonlinear compensation system,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper W3F.8.

Ip, E.

Kahn, J. M.

Karar, A. S.

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

Kim, I.

Kumar, S.

X. Liang and S. Kumar, “Optical back propagation for compensating nonlinear impairments in fiber optic links with ROADMs,” Opt. Express 24(20), 22682–22692 (2016).
[Crossref] [PubMed]

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]

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, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (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]

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 and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
[Crossref] [PubMed]

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

S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett. 13(8), 800–802 (2001).
[Crossref]

S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley, 2014).
[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. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.6.

Li, C.

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[Crossref]

Li, G.

Li, L.

Li, X.

Liang, X.

Liu, L.

Lowery, A. J.

B. Foo, B. Corcoran, C. Zhu, and A. J. Lowery, “Distributed nonlinear compensation of dual-polarization signals using optoelectronics,” IEEE Photon. Technol. Lett. 28(20), 2141–2144 (2016).
[Crossref]

M. Morshed, L. B. Du, and A. J. Lowery, “Mid-span spectral inversion for coherent optical OFDM systems: fundamental limits to performance,” J. Lightwave Technol. 31(1), 58–66 (2013).
[Crossref]

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 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.

McCarthy, M.

A. D. Ellis and M. McCarthy, “Impact of optical phase conjugation on the Shannon capacity limit,” in Optical Fiber Communication Conference, (Optical Society of America, 2016), paper Th4F.2.

Mecozzi, A.

A. Mecozzi, C.B. Clausen, and M. Shtaif, “System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(12), 1633–1635 (2000).
[Crossref]

Menyuk, C. R.

C. R. Menyuk, “Interaction of nonlinearity and polarization mode dispersion,” J. Opt. Fiber Commun. Rep. 1(4), 305–311 (2004).
[Crossref]

Mikkelsen, B.

R.J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” IEEE Electron. Lett. 35(18), 1576–1578 (1999).
[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 Photon. Technol. Lett. 18(9), 995–997 (2006).
[Crossref]

Mishra, S. K.

S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).

Mollenauer, L. F.

Morshed, M.

M. Morshed, L. B. Du, and A. J. Lowery, “Mid-span spectral inversion for coherent optical OFDM systems: fundamental limits to performance,” J. Lightwave Technol. 31(1), 58–66 (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 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Namiki, S.

K. Solis-Trapala, T. Inoue, and S. Namiki, “Nearly-ideal optical phase conjugation based nonlinear compensation system,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper W3F.8.

Noé, R.

O’Sullivan, M.

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[Crossref]

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in 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 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

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

Pepper, D.M.

Pfau, T.

Plant, D. V.

Poggiolini, P.

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 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

Raybon, G.

R.J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” IEEE Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

Roberts, K.

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[Crossref]

Y. Gao, A. S. Karar, J. C. Cartledge, S. S.-H. Yam, M. O’Sullivan, C. Laperle, A. Borowiec, and K. Roberts, “Simplified nonlinearity pre-compensation using a modified summation criteria and non-uniform power profile,” in 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]

Shtaif, M.

A. Mecozzi, C.B. Clausen, and M. Shtaif, “System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(12), 1633–1635 (2000).
[Crossref]

Solis-Trapala, K.

K. Solis-Trapala, T. Inoue, and S. Namiki, “Nearly-ideal optical phase conjugation based nonlinear compensation system,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper W3F.8.

Srikant, V.

S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).

Stone, J. S.

S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).

Strawczynski, L.

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[Crossref]

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]

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 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

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]

Winzer, P. J.

R. J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in European Conference on Optical Communication (ECOC 2005), paper Tu 3.2.2.

Yam, S. S.-H.

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

Yaman, F.

Yamauchi, 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Tu3A.3.

Yan, W.

Yang, D.

Yariv, A.

Zhu, C.

B. Foo, B. Corcoran, C. Zhu, and A. J. Lowery, “Distributed nonlinear compensation of dual-polarization signals using optoelectronics,” IEEE Photon. Technol. Lett. 28(20), 2141–2144 (2016).
[Crossref]

Zhu, L.

IEEE Electron. Lett. (1)

R.J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems,” IEEE Electron. Lett. 35(18), 1576–1578 (1999).
[Crossref]

IEEE Photon. Technol. Lett. (8)

A. Mecozzi, C.B. Clausen, and M. Shtaif, “System impact of intra-channel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12(12), 1633–1635 (2000).
[Crossref]

S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett. 13(8), 800–802 (2001).
[Crossref]

K. Roberts, C. Li, L. Strawczynski, M. O’Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[Crossref]

J. Shao, S. Kumar, and X. Liang, “Digital back propagation with optimal step size for polarization multiplexed transmission,” IEEE Photon. Technol. Lett. 25(23), 2327–2330 (2013).
[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 Photon. 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 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]

B. Foo, B. Corcoran, C. Zhu, and A. J. Lowery, “Distributed nonlinear compensation of dual-polarization signals using optoelectronics,” IEEE Photon. Technol. Lett. 28(20), 2141–2144 (2016).
[Crossref]

J. Lightwave Technol. (7)

J. Opt. Fiber Commun. Rep. (1)

C. R. Menyuk, “Interaction of nonlinearity and polarization mode dispersion,” J. Opt. Fiber Commun. Rep. 1(4), 305–311 (2004).
[Crossref]

Opt. Express (9)

S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007).
[Crossref] [PubMed]

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]

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]

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, “Correlated digital back propagation based on perturbation theory,” Opt. Express 23(11), 14655–14665 (2015).
[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. Express 16(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. Express 16(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. Express 18(16), 17075–17088 (2010).
[Crossref] [PubMed]

X. Liang and S. Kumar, “Optical back propagation for compensating nonlinear impairments in fiber optic links with ROADMs,” Opt. Express 24(20), 22682–22692 (2016).
[Crossref] [PubMed]

Opt. Lett. (5)

Other (10)

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.

A. D. Ellis and M. McCarthy, “Impact of optical phase conjugation on the Shannon capacity limit,” in Optical Fiber Communication Conference, (Optical Society of America, 2016), paper Th4F.2.

K. Solis-Trapala, T. Inoue, and S. Namiki, “Nearly-ideal optical phase conjugation based nonlinear compensation system,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper W3F.8.

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

R. J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in European Conference on Optical Communication (ECOC 2005), paper Tu 3.2.2.

S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (Wiley, 2014).
[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 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 Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th2A.8.

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

S. R. Bickham, M. B. Cain, S. Kumar, S. K. Mishra, V. Srikant, and J. S. Stone, “Dispersion slope compensating optical fiber,” U.S. patent application6,671,445 (December30, 2003).

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Figures (10)

Fig. 1
Fig. 1 Schematics of back propagation based on (a) a virtual fiber and (b) a DVF. Tx: transmitter, Rx: receiver, TF: transmission fiber, OPC: optical phase conjugator, DVF: dispersion varying fiber.
Fig. 2
Fig. 2 Dispersion profiles of optical back propagation fibers for the case of full EDFA amplification. G′ is the gain of the pre-amplifier.
Fig. 3
Fig. 3 Dispersion profiles of optical back propagation fibers for the case of hybrid amplification (EDFA gain = 4.8 dB, Raman gain = 7.2 dB). G′ is the gain of the pre-amplifier.
Fig. 4
Fig. 4 Dispersion profiles of optical back propagation fibers for the case of full Raman amplification. G′ is the gain of the pre-amplifier.
Fig. 5
Fig. 5 DVF 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 0.02.
Fig. 6
Fig. 6 (a) Fiber optic system with an OBP module at each network node. (b) OBP module for multi-channel systems. Tx: transmitter, Rx: receiver, TF: transmission fiber, OBP: optical back propagation, ROADM: reconfigurable optical add-drop multiplexer, DeMUX: demultiplexer, BPF: band pass filter, MUX: multiplexer, OPC: optical phase conjugator, DVF: dispersion varying fiber, Att.: attenuator.
Fig. 7
Fig. 7 Q-factor vs. launch power per WDM channel. Transmission distance = 3000 km, modulation = 64 QAM.
Fig. 8
Fig. 8 (a) Signal power profile in transmission fiber 1 (TF1) before an OPC (blue line). (b) Signal power profile in transmission fiber 2 (TF2) after an OPC (green line), and the desired power profile in TF2 (red line) that is symmetric to the power profile in TF1.
Fig. 9
Fig. 9 Comparison of OBP with OPC only case. The OPC only case has an OPC at each node along the transmission link. Transmission distance = 3000 km, modulation = 64 QAM.
Fig. 10
Fig. 10 Comparisons of different nonlinear compensation schemes for a point-to-point system. The WDM channels are not added/dropped throughout the fiber optic link. Transmission distance = 3000 km, modulation = 64 QAM.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

q ( t , z ) z + α ( z ) 2 q ( t , z ) + i β 2 2 2 q ( t , z ) t 2 = i γ | q ( t , z ) | 2 q ( t , z ) ,
D ( t ) = β 2 2 2 t 2 , N ( t , z ) = γ | q ( t , z ) | 2 + i α ( z ) 2 .
q ( t , L ) = M q ( t , 0 ) ,
M = exp { i 0 L [ D ( t ) + N ( t , z ) ] d z } ,
q o u t ( t ) = M 1 q ( t , L ) = q ( t , 0 ) .
M 1 = exp { i 0 L [ D ( t ) + N ( t , L z ) ] d z } .
q o u t ( t ) = M q ( t , L ) = q ( t , 0 ) ,
M = exp { i 0 L [ D ( t ) + N ( t , L z ) ] d z } .
α ( z ) = α ( L z ) , β 2 = β 2 , γ = γ .
q b z b + α ( z b ) 2 q b + i β 2 2 2 q b t 2 = i γ | q b | 2 q b ,
q b ( t , z b ) = P b e W ( z b ) / 2 u b ,
d z b = β 2 d z b ,
u b z b + i 2 2 u b t 2 = i γ P b e W ( z b ) β 2 | u b | 2 u b ,
q b z d + α d 2 q b + i β 2 , d ( z d ) 2 2 q b t 2 = i γ d | q b | 2 q b ,
q b ( t , 0 ) = G q ( t , L ) .
q b ( t , z b ) = P d e α d z d / 2 u b ,
d z d = β 2 , d ( z d ) d z d ,
u b z d + i 2 2 u b t 2 = i γ d P d e α d z d β 2 , d ( z d ) | u b | 2 u b .
γ e W ( z b ) β 2 = G γ d e α d z d β 2 , d ,
d z b = d z d .
β 2 β 2 , d ( z d ) = d z d d z b .
0 z b e W ( z b ) d z b = G γ d ( 1 e α d z d ) γ α d .
A ( x ) = 0 x e W ( x ) d x .
A ( z b ) = G γ d ( 1 e α d z d ) γ α d .
β 2 , d ( z d ) = G β 2 γ d γ e W ( z b ) α d z d ,
z b = A 1 [ G γ d ( 1 e α d z d ) γ α d ] .
L d = 1 α d l n [ 1 α d γ A ( L ) G γ d ] .
α ( z ) = α s K e α p z ,
K = g R P p e α p L A p ,
β 2 , d ( z d ) = e α d z d γ γ d G + α s ( 1 e α d z d α d ) β 2 ,
L d = 1 α d l n [ 1 α d γ G α s γ d ( e α s L 1 ) ] ,
β ^ 2 , d ( z ) = [ 1 + x ( z ) ] β 2 , d ( z ) ,

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