Periodic nonlinear Fourier transform for fiber optic communications Part I Theory and numerical methods

In this work, we introduce the periodic nonlinear Fourier transform (PNFT) method as an alternative and efficacious tool for compensation of the nonlinear transmission effects in optical fiber links. In the Part I, we introduce the algorithmic platform of the technique, describing in details the direct and inverse PNFT operations, also known as the inverse scattering transform for periodic (in time variable) nonlinear Schrodinger equation (NLSE) We pay a special attention to explaining the potential advantages of the PNFT based processing over the previously studied nonlinear Fourier transform (NFT) based methods. Further, we elucidate the issue of the numerical PNFT computation: we compare the performance of four known numerical methods applicable for the calculation of nonlinear spectral data (the direct PNFT) in particular, taking the main spectrum (utilized further in Part II for the modulation and transmission) associated with some simple example waveforms as the quality indicator for each method. We show that the Ablowitz Ladik discretization approach for the direct PNFT provides the best performance in terms of the accuracy and computational time consumption.  Scaling optical fiber networks: challenges and solutions," Opt. Photon. News 26, 28 35 (2015) 5. E. Ip and J. M. Kahn "Compensation of dispersion and nonlinear impairments using digital backpropagation,"  (2008) 6. X. Chen, X. Liu, S. Chandrasekhar, B. Zhu, and R. W. Tkach, "Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation," in Technical Digest of Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC) paper OTh3C.1 (2012) 7. S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, W. Sohler, G. D. Khoe, and H. de Waardt, "Optical phase conjugation for ultra long haul phase shift keyed transmission,"  (2006) 8. I. Phillips, M. Tan, M. F. Stephens, M. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, "Exceeding the nonlinear shannon limit using Raman laser based amplification and optical phase conjugation," in Optical Fiber Communication Conference and Exhibition (OFC) Los Angeles, USA, paper M3C.1 (2014) 9. A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, "Phase conjugated twin waves for communication beyond the Kerr nonlinearity limit,"  (2013) 10. D. Rafique, "Fiber nonlinearity compensation: commercial applications and complexity analysis,"  (2016) 11. V. E. Zakharov and A. B. Shabat, "Exact theory of two dimensional self focusing and one dimensional selfmodulation of waves in nonlinear media," Soviet Physics JETP 34, 62 69 (1972) 12. M. I. Yousefi and F. R. Kschischang, "Information transmission using the nonlinear Fourier transform, part I: mathematical tools,"  (2014) 13. M. I. Yousefi and F. R. Kschischang, "Information transmission using the nonlinear Fourier transform, part II: numerical methods,"  (2014) 14. M. I. Yousefi and F. R. Kschischang, "Information transmission using the nonlinear Fourier transform, part III: spectrum modulation,"  4346 4369 (2014) 15. J. E. Prilepsky and S. K. Turitsyn, "Eigenvalue communications in nonlinear fiber channels," in Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, Ch. 18, edts. K. Porsezian and R. Ganapathy, pp. 459 490 (CRC, 2015) 16. A. Maruta, Y. Matsuda, H. Terauchi, and A. Toyota, "Digital coherent technology based eigenvalue modulated optical fiber transmission system," in Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, Ch. 19, edts. K. Porsezian and R. Ganapathy, pp. 491 506 (CRC, 2015) 17. A. Hasegawa and T. Nyu, "Eigenvalue communication,"  (1993) 18. S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, "Nonlinear inverse synthesis technique for optical links with lumped amplification," Opt. Express 23, 8317 8328 (2015) 19. A. Hasegawa and Y. Kodama, "Guiding center soliton,"  (1991) 20. E. G. Turitsyna and S. K. Turitsyn, "Digital signal processing based on inverse scattering transform." Opt. Lett. 38, 4186 4188 (2013) 21. J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, "Nonlinear spectral management: linearization of the lossless fiber channel," Opt. Express 21, 24344 24367 (2013) 22. J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, "Nonlinear Inverse Synthesis and Eigenvalue Division Multiplexing in Optical Fiber Channels."  (2014) 23. S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, "Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers," Opt. Express 22, 26720 26741 (2014) 24. H. Bulow, "Experimental demonstration of optical signal detection using nonlinear fourier transform."  (2015) 25. V. Aref, H. BAijlow, K. Schuh, and W. Idler, "Experimental demonstration of nonlinear frequency division multiplexed transmission," in European Conference on Optical Communication (ECOC) Valencia, Spain (2015) 26. H. Bulow, V. Aref, and W. Idler, "Transmission of waveforms determined by 7 eigenvalues with PSK modulated spectral amplitudes,"  (2016) 27. M. I. Yousefi and X. Yangzhang, "Linear and nonlinear frequency division multiplexing,"  (2016) 28. Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. A. Wai, F. R. Kschischang, and A. Lau, "Nonlinear frequency division multiplexed transmissions based on NFT,"  (2015) 29. A. Geisler and C. G. Schaeffer, "Implementation of eigenvalue multiplex transmission with a real Fiber Link using the discrete nonlinear Fourier spectrum,"  (2016) Leipzig, Germany, pp. 1 6. 30. A. Maruta, "Eigenvalue modulated optical transmission system (invited)" in The 20th OptoElectronics and Communications Conference (OECC) Shanghai, China, Paper JThA.21 (2015) 31. S. Turitsyn, "Nonlinear Fourier transform for optical communications," in Frontiers in Optics 2015, OSA Technical Digest (online) paper FM1E.6 (2015) 32. S. Hari, F. Kschischang, and M. Yousefi, "Multi eigenvalue communication via the nonlinear Fourier transform," 27th Biennial Symposium on Communications (QBSC) pp. 92 95 (2014) 33. S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, and S. K. Turitsyn, "Digital backpropagation in the nonlinear Fourier domain,"  Stockholm, Sweden, pp. 445 449 (2015) 34. S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania CastaAsAsn, P. Harper, and S. K. Turitsyn "Modified nonlinear inverse synthesis for optical links with distributed Raman amplification," in European Conference on Optical Communication (ECOC) Valencia, Spain, paper Tu.1.1.3 (2015) 35. A. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, 1st ed.