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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 four-wave 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 few-mode 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.
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Keywords:
- space division multiplexing /
- few mode fiber /
- four-wave mixing /
- analytic solution
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图 1 非简并FWM过程中导波光的模式与频谱分布
Fig. 1. Mode and frequency distribution in the non-degenerate FWM process.
图 2 相位失配因子对
$\Delta {f_1}$ 和$\Delta {f_2}$ 的依赖Fig. 2. Dependency of phase mismatching factor on
$\Delta {f_1}$ and$\Delta {f_2}$ .
图 3 解析解与数值结果的比较 (a)闲频光功率; (b)闲频光相位
Fig. 3. Comparison of analytical and numerical results: (a) Idler output power; (b) idler output phase.
图 4 (a) FWM转移能量在不同相位失配
$\Delta \beta $ 随光纤长度L的变化; (b) 相位匹配度$\mu $ 随$\Delta \beta $ 的变化Fig. 4. (a) FWM energy transfer
$q(z)$ of different$\Delta \beta $ with fiber length L; (b) variation of phase matching parameter$\mu $ with$\Delta \beta $ .
图 5 输出闲频光相位
${\varphi _4}$ 与输入初相位运算${\varphi _{10}} + $ ${\varphi _{20}} - {\varphi _{30}}$ 之间的关系Fig. 5. Dependence of output idler phase
${\varphi _4}$ on the initial phase operation of${\varphi _{10}} + {\varphi _{20}} - {\varphi _{30}}$ .
图 7 基于FWM解析解的非线性补偿框图
Fig. 7. Block diagram of nonlinear compensation based on the analytic solution for FWM effect.
表 1 模场的归一化交叠积分参数
Table 1. Normalized overlap integral parameters of mode fields.
模场分布 flmnp/fLP01/arb.units 4束光都为LP01模 1.000 4束光都为LP11a或LP11b模 0.747 2束光为LP01模、2束光为LP11a或LP11b模 0.496 2束光为LP11a模、2束光为LP11b模 0.249 
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[1] Ellis A D, Zhao J, Cotter D 2010 J. Lightwave Technol. 28 423
Google Scholar
[2] Ellis A D, McCarthy M E, Khateeb M A Z A, Sorokina M, Doran N J 2017 Adv. Opt. Photonics 9 429
Google Scholar
[3] Richardson D J, Fini J M, Nelson L E 2013 Nat. Photonics 7 354
Google Scholar
[4] Li G, Bai N, Zhao N, Xia C 2014 Adv. Opt. Photonics 6 413
Google Scholar
[5] Mizuno T, Takara H, Sano A, Miyamoto Y 2016 J. Lightwave Technol. 34 582
Google Scholar
[6] 郑兴娟, 任国斌, 黄琳, 郑鹤玲 2016 物理学报 65 064208
Google Scholar
Zheng X J, Ren G B, Huang L, Zheng H L 2016 Acta Phys. Sin. 65 064208
Google Scholar
[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
Google Scholar
[8] Kitayama K, Diamantopoulos N 2017 IEEE Commun. Mag. 55 163
[9] Nazemosadat E, Lorences-Riesgo A, Karlsson M, Andrekson P A 2017 J. Lightwave Technol. 35 2810
Google Scholar
[10] 姚殊畅, 付松年, 张敏明, 唐明, 沈平, 刘德明 2013 物理学报 62 144215
Google Scholar
Yao S C, Fu S N, Zhang M M, Tang M, Shen P, Liu D M 2013 Acta Phys. Sin. 62 144215
Google Scholar
[11] Essiambre R J, Kramer G, Winzer P J, Foschini G J, Goebel B 2010 J. Lightwave Technol. 28 662
Google Scholar
[12] Mumtaz S, Essiambre R J, Agrawal G P 2013 J. Lightwave Technol. 31 398
Google Scholar
[13] Suibhne N M, Ellis A D, Gunning F C G, Sygletos S 2013 39th 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 R W, Chraplyvy A R, Sun Y, Jiang X, Lingle Jr R 2013 IEEE Photonics Technol. Lett. 25 539
Google Scholar
[15] Rademacher G, Petermann K 2016 J. Lightwave Technol. 34 2280
Google Scholar
[16] Trichili A, Zghal M, Palmieri L, Santagiustina M 2017 IEEE Photonics J. 9 1
[17] Marhic M E 2013 J. Opt. Soc. Am. B 30 62
Google Scholar
[18] 曹亚敏, 武保剑, 万峰, 邱昆 2018 物理学报 67 094208
Google Scholar
Cao Y M, Wu B J, Wan F, Qiu K 2018 Acta Phys. Sin. 67 094208
Google Scholar
[19] Poletti F, Horak P 2008 J. Opt. Soc. Am. B 25 1645
Google Scholar
[20] Ferreira F, Jansen S, Monteiro P, Silva H 2012 IEEE Photonics Technol. Lett. 24 240
Google Scholar
[21] Antonelli C, Shtaif M, Mecozzi A 2016 J. Lightwave Technol. 34 36
Google Scholar
[22] Rademacher G, Warm S, Petermann K 2012 IEEE Photonics Technol. Lett. 24 1929
Google Scholar
[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
Google Scholar
[25] Brehler M, Schirwon M, Göddeke D, Krummrich P M 2017 J. Lightwave Technol. 35 3622
Google Scholar
[26] Hu X, Wang A, Zeng M, Long Y, Zhu L, Fu L, Wang J 2016 Sci. Rep. 6 32911
Google Scholar
[27] Wang A, Hu X, Zhu L, Zeng M, Fu L, Wang J 2015 Opt. Express 23 31728
Google Scholar
[28] Gui C, Wang J 2014 Sci. Rep. 4 7491
[29] Wang J, Yang J Y, Huang H, Willner A E 2013 Opt. Express 21 488
Google Scholar
[30] Wang J, Yang J, Wu X, Willner A E 2012 J. Lightwave Technol. 30 2890
Google Scholar
[31] Wang J, Nuccio S R, Yang J Y, Wu X, Bogoni A, Willner A E 2012 Opt. Lett. 37 1139
Google Scholar
[32] Tsang M, Psaltis D, Omenetto F G 2003 Opt. Lett. 28 1873
Google Scholar
[33] Mateo E F, Zhou X, Li G 2011 Opt. Express 19 570
Google Scholar
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