空频复用光纤中四波混频过程的解析分析方法
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摘要
空分复用(SDM)与波分复用(WDM)的结合有效提升了光纤通信系统的容量,其中光纤的非线性影响不可忽视.本文将光纤四波混频作用过程由WDM频域推广到模分复用(MDM)空间域,第一次给出抽运消耗情形下空频域四波混频的统一解析表达式.通过数值计算非简并四波混频耦合模方程的幅度和相位演化特性,验证了解析分析方法的正确性.讨论了解析解在多波耦合方程简化、大规模并行相位运算器设计以及快速非线性补偿等方面的应用.
In recent years, the transmission capacity of wavelength division multiplexing(WDM) communication systems has gradually approached to the nonlinear Shannon limit. To meet the increasing demand for communication capacity, space division multiplexing(SDM) has become one of the most concerned technologies.In this paper, the four-wave mixing process(FWM) in fibers is considered from the frequency domain to the mode division multiplexing(MDM) spatial domain under pump depletion and the exact analytical solution to the FWM coupled-mode equations in the space-frequency domain is in detail deduced. The analytical method is verified by numerically calculating the amplitude and phase evolution of the idler wave in non-degenerate fourwave mixing. We discuss three new applications of the analytical solution as follows. 1) Using the phase matching condition we select the terms in the multi-wave coupling equation, and only retain the coupling term that plays a major role. According to the analytical solution in this paper, the phase matching percentage parameter is introduced to determine the FWM coupling terms necessary for multi-wave coupling equations,thus simplifying the multi-wave coupling problem in the study. 2) Combining the analytical solution with the numerical calculation results, we find the initial phase relationship between the output idler and the input guided wave for phase-insensitive FWM, and we provide the analytical expression for a theoretical basis to efficiently design the FWM-based phase arithmetic devices in parallel operating at WDM and MDM systems.3) We propose a nonlinear compensation algorithm based on analytical solution, which can be used in the fewmode transmission system. The algorithm can fast evaluate or compensate for the fiber nonlinearity by taking into account the pump depletion of the FWM effect. Compared with the traditional digital back propagation(DBP) algorithm, our algorithm has the advantage of lower complexity.
In recent years, the transmission capacity of wavelength division multiplexing(WDM) communication systems has gradually approached to the nonlinear Shannon limit. To meet the increasing demand for communication capacity, space division multiplexing(SDM) has become one of the most concerned technologies.In this paper, the four-wave mixing process(FWM) in fibers is considered from the frequency domain to the mode division multiplexing(MDM) spatial domain under pump depletion and the exact analytical solution to the FWM coupled-mode equations in the space-frequency domain is in detail deduced. The analytical method is verified by numerically calculating the amplitude and phase evolution of the idler wave in non-degenerate fourwave mixing. We discuss three new applications of the analytical solution as follows. 1) Using the phase matching condition we select the terms in the multi-wave coupling equation, and only retain the coupling term that plays a major role. According to the analytical solution in this paper, the phase matching percentage parameter is introduced to determine the FWM coupling terms necessary for multi-wave coupling equations,thus simplifying the multi-wave coupling problem in the study. 2) Combining the analytical solution with the numerical calculation results, we find the initial phase relationship between the output idler and the input guided wave for phase-insensitive FWM, and we provide the analytical expression for a theoretical basis to efficiently design the FWM-based phase arithmetic devices in parallel operating at WDM and MDM systems.3) We propose a nonlinear compensation algorithm based on analytical solution, which can be used in the fewmode transmission system. The algorithm can fast evaluate or compensate for the fiber nonlinearity by taking into account the pump depletion of the FWM effect. Compared with the traditional digital back propagation(DBP) algorithm, our algorithm has the advantage of lower complexity.
引文
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[3]Richardson D J,Fini J M,Nelson L E 2013 Nat.Photonics 7354
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[5]Mizuno T,Takara H,Sano A,Miyamoto Y 2016 J.Lightwave Technol.34 582
[6]Zheng X J,Ren G B,Huang L,Zheng H L 2016 Acta Phys.Sin.65 064208(in Chinese)[郑兴娟,任国斌,黄琳,郑鹤玲2016物理学报65 064208]
[7]Rademacher G,Ryf R,Fontaine N K,Chen H,Essiambre R,Puttnam B J,Luís R S,Awaji Y,Wada N,Gross S,Riesen N,Withford M,Sun Y,Lingle R 2018 J.Lightwave Technol.36 1382
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[9]Nazemosadat E,Lorences-Riesgo A,Karlsson M,Andrekson P A 2017 J.Lightwave Technol.35 2810
[10]Yao S C,Fu S N,Zhang M M,Tang M,Shen P,Liu D M2013 Acta Phys.Sin.62 144215(in Chinese)[姚殊畅,付松年,张敏明,唐明,沈平,刘德明2013物理学报62 144215]
[11]Essiambre R J,Kramer G,Winzer P J,Foschini G J,Goebel B 2010 J.Lightwave Technol.28 662
[12]Mumtaz S,Essiambre R J,Agrawal G P 2013 J.Lightwave Technol.31 398
[13]Suibhne N M,Ellis A D,Gunning F C G,Sygletos S 201339th European Conference and Exhibition on Optical Communication London,UK,September 22-26,2013 p882
[14]Essiambre R J,Mestre M A,Ryf R,Gnauck A H,Tkach RW,Chraplyvy A R,Sun Y,Jiang X,Lingle Jr R 2013 IEEEPhotonics Technol.Lett.25 539
[15]Rademacher G,Petermann K 2016 J.Lightwave Technol.342280
[16]Trichili A,Zghal M,Palmieri L,Santagiustina M 2017 IEEEPhotonics J.9 1
[17]Marhic M E 2013 J.Opt.Soc.Am.B 30 62
[18]Cao Y M,Wu B J,Wan F,Qiu K 2018 Acta Phys.Sin.67094208(in Chinese)[曹亚敏,武保剑,万峰,邱昆2018物理学报67 094208]
[19]Poletti F,Horak P 2008 J.Opt.Soc.Am.B 25 1645
[20]Ferreira F,Jansen S,Monteiro P,Silva H 2012 IEEEPhotonics Technol.Lett.24 240
[21]Antonelli C,Shtaif M,Mecozzi A 2016 J.Lightwave Technol.34 36
[22]Rademacher G,Warm S,Petermann K 2012 IEEE Photonics Technol.Lett.24 1929
[23]Agrawal G P 2005 Nonlinear Fiber Optics(New York:Academic Press)pp195-211
[24]Xiao Y,Essiambre R J,Desgroseilliers M,Tulino A M,Ryf R,Mumtaz S,Agrawal G P 2014 Opt.Express 22 32039
[25]Brehler M,Schirwon M,G?ddeke D,Krummrich P M 2017 J.Lightwave Technol.35 3622
[26]Hu X,Wang A,Zeng M,Long Y,Zhu L,Fu L,Wang J 2016Sci.Rep.6 32911
[27]Wang A,Hu X,Zhu L,Zeng M,Fu L,Wang J 2015 Opt.Express 23 31728
[28]Gui C,Wang J 2014 Sci.Rep.4 7491
[29]Wang J,Yang J Y,Huang H,Willner A E 2013 Opt.Express21 488
[30]Wang J,Yang J,Wu X,Willner A E 2012 J.Lightwave Technol.30 2890
[31]Wang J,Nuccio S R,Yang J Y,Wu X,Bogoni A,Willner AE 2012 Opt.Lett.37 1139
[32]Tsang M,Psaltis D,Omenetto F G 2003 Opt.Lett.28 1873
[33]Mateo E F,Zhou X,Li G 2011 Opt.Express 19 570