{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T20:47:51Z","timestamp":1768337271196,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T00:00:00Z","timestamp":1590710400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["17-72-30006"],"award-info":[{"award-number":["17-72-30006"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003443","name":"Ministry of Education and Science of the Russian Federation","doi-asserted-by":"publisher","award":["0305-2019-0006"],"award-info":[{"award-number":["0305-2019-0006"]}],"id":[{"id":"10.13039\/501100003443","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the present paper we summarize the methods and results of calculations for the theoretical informational quantities obtained in our works for the nondispersive optical fiber channel. We considered two models: the per-sample model and the model where the input signal depends on time. For these models we found the approach for the calculation of the mutual information exactly in the nonlinearity parameter but for the large signal-to-noise power ratio. Using this approach for the per-sample model we found the lower bound of the channel capacity in the intermediate power range.<\/jats:p>","DOI":"10.3390\/e22060607","type":"journal-article","created":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T11:49:21Z","timestamp":1591012161000},"page":"607","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Path Integral Approach to Nondispersive Optical Fiber Communication Channel"],"prefix":"10.3390","volume":"22","author":[{"given":"Aleksei V.","family":"Reznichenko","sequence":"first","affiliation":[{"name":"Theoretical Department, Budker Institute of Nuclear Physics of Siberian Branch Russian Academy of Sciences, 630090 Novosibirsk, Russia"},{"name":"Department of Physics, Novosibirsk State University, 630090 Novosibirsk, Russia"}]},{"given":"Ivan S.","family":"Terekhov","sequence":"additional","affiliation":[{"name":"Theoretical Department, Budker Institute of Nuclear Physics of Siberian Branch Russian Academy of Sciences, 630090 Novosibirsk, Russia"},{"name":"Department of Physics, Novosibirsk State University, 630090 Novosibirsk, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ilyas, M., and Mouftah, H.T. (2003). The Handbook of Optical Communication Networks, CRC Press LLC.","DOI":"10.1201\/9780203489642"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1122","DOI":"10.1364\/JOSAB.8.001122","article-title":"Quantum noise in a solitonlike repeater system","volume":"8","author":"Haus","year":"1991","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1109\/50.336065","article-title":"Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers","volume":"12","author":"Mecozzi","year":"1994","journal-title":"J. Light. Technol."},{"key":"ref_5","unstructured":"Iannoe, E., Matera, F., Mecozzi, A., and Settembre, M. (1998). Nonlinear Optical Communication Networks, John Wiley & Sons."},{"key":"ref_6","first-page":"62","article-title":"Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media","volume":"34","author":"Zakharov","year":"1972","journal-title":"Soviet J. Exp. Theory Phys."},{"key":"ref_7","unstructured":"Novikov, S., Manakov, S.V., Pitaevskii, L.P., and Zakharov, V.E. (1984). Theory of Solitons: The Inverse Scattering Method (Monographs in Contemporary Mathematics), Springer."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"163901","DOI":"10.1103\/PhysRevLett.101.163901","article-title":"Capacity Limits of Information Transport in Fiber-Optic Networks","volume":"101","author":"Essiambre","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1109\/JLT.2009.2039464","article-title":"Capacity Limits of Optical Fiber Networks","volume":"28","author":"Essiambre","year":"2010","journal-title":"J. Light. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3861","DOI":"10.1038\/ncomms4861","article-title":"Regeneration limit of classical Shannon capacity","volume":"5","author":"Sorokina","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"230602","DOI":"10.1103\/PhysRevLett.113.230602","article-title":"Conditional Probability Calculations for the Nonlinear Schr\u00f6dinger Equation with Additive Noise","volume":"113","author":"Terekhov","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"30053","DOI":"10.1364\/OE.22.030053","article-title":"On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission","volume":"22","author":"Liga","year":"2014","journal-title":"Opt. Express"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1364\/OL.42.000121","article-title":"Achievable information rates estimates in optically amplified transmission systems using nonlinearity compensation and probabilistic shaping","volume":"42","author":"Semrau","year":"2017","journal-title":"Opt. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5131","DOI":"10.1109\/TIT.2018.2791592","article-title":"Autocorrelation Function for Dispersion-Free Fiber Channels with Distributed Amplification","volume":"64","author":"Kramer","year":"2018","journal-title":"IEEE Tran. Inf. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1104","DOI":"10.1109\/50.939789","article-title":"The Shannon channel capacity of dispersion-free nonlinear optical fiber transmission","volume":"19","author":"Tang","year":"2001","journal-title":"J. Light. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1110","DOI":"10.1109\/50.939790","article-title":"The multispan effects of Kerr nonlinearity and amplifier noises on Shannon channel capacity of a dispersion-free nonlinear optical fiber","volume":"19","author":"Tang","year":"2001","journal-title":"J. Light. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"203901","DOI":"10.1103\/PhysRevLett.91.203901","article-title":"Information Capacity of Optical Fiber Channels with Zero Average Dispersion","volume":"91","author":"Turitsyn","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"7522","DOI":"10.1109\/TIT.2011.2165793","article-title":"On the Per-Sample Capacity of Nondispersive Optical Fibers","volume":"57","author":"Yousefi","year":"2011","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"062133","DOI":"10.1103\/PhysRevE.95.062133","article-title":"The loglog growth of channel capacity for nondispersive nonlinear optical fiber channel in intermediate power range","volume":"95","author":"Terekhov","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"012127","DOI":"10.1103\/PhysRevE.95.012127","article-title":"Next-to-leading order corrections to capacity for nondispersive nonlinear optical fiber channel in intermediate power region","volume":"95","author":"Panarin","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"012133","DOI":"10.1103\/PhysRevE.99.012133","article-title":"The loglog growth of channel capacity for nondispersive nonlinear optical fiber channel in intermediate power range. Extension of the model","volume":"99","author":"Reznichenko","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Keykhosravi, K., Durisi, G., and Agrell, E. (2019). Accuracy Assessment of Nondispersive Optical Perturbative Models Through Capacity Analysis. Entropy, 21.","DOI":"10.3390\/e21080760"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1038\/35082518","article-title":"Nonlinear limits to the information capacity of optical fibre communications","volume":"411","author":"Mitra","year":"2001","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6343","DOI":"10.1103\/PhysRevE.60.6343","article-title":"Effective action for stochastic partial differential equations","volume":"60","author":"Hochberg","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Gardiner, C.W. (1985). Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer.","DOI":"10.1007\/978-3-662-02452-2"},{"key":"ref_26","unstructured":"Feynman, R.P., and Hibbs, A.R. (1965). Quantum Mechanics and Path Integrals, McGraw-Hill."},{"key":"ref_27","unstructured":"Lavrentiev, M.A., and Shabat, B.V. (1987). Method of Complex Function Theory, Nauka."},{"key":"ref_28","unstructured":"Gradshtein, I.S., and Ryzik, I.M. (2014). Table of Integrals and Series, and Products, Academic Press."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"042203","DOI":"10.1103\/PhysRevE.94.042203","article-title":"Calculation of mutual information for nonlinear communication channel at large SNR","volume":"94","author":"Terekhov","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_30","first-page":"186","article-title":"Channel Capacity and Simple Correlators for Nonlinear Communication Channel at Large SNR and Small Dispersion","volume":"2018","author":"Reznichenko","year":"2018","journal-title":"IEEE Int. Symp. Inf. Theory Proc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"012026","DOI":"10.1088\/1742-6596\/826\/1\/012026","article-title":"Calculation of mutual information for nonlinear optical-fiber communication channel at large SNR within path-integral formalism","volume":"826","author":"Reznichenko","year":"2017","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"012016","DOI":"10.1088\/1742-6596\/999\/1\/012016","article-title":"Channel Capacity Calculation at Large SNR and Small Dispersion within Path-Integral Approach","volume":"999","author":"Reznichenko","year":"2018","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"012013","DOI":"10.1088\/1742-6596\/1206\/1\/012013","article-title":"Investigation of Nonlinear Communication Channel with Small Dispersion via Stochastic Correlator Approach","volume":"1206","author":"Reznichenko","year":"2019","journal-title":"J. Phys. Conf. 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