空频复用光纤中四波混频过程的解析分析方法

万峰; 武保剑; 曹亚敏; 王瑜浩; 文峰; 邱昆

doi:10.7498/aps.68.20182129

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.

关键词:

Abstract

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.

Keywords:

PACS:

作者及机构信息

电子科技大学信息与通信工程学院, 光纤传感与通信教育部重点实验室, 成都　611731

通信作者: 武保剑, bjwu@uestc.edu.cn

基金项目: 国家自然科学基金(批准号: 61671108, 61505021)资助的课题.

Authors and contacts

Key Laboratory of Optical Fiber Sensing and Communication, Ministry of Education, School of Information and Communication Engineering, University of Electonic Science and Technology of China, Chengdu 611731, China

Corresponding author: Wu Bao-Jian, bjwu@uestc.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61671108, 61505021).

文章全文

参考文献

[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
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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.

下载: 全尺寸图片幻灯片

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 $\Delta {f_2}$ .

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

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 $\Delta \beta$ .

下载: 全尺寸图片幻灯片

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 ${\varphi _{10}} + {\varphi _{20}} - {\varphi _{30}}$ .

下载: 全尺寸图片幻灯片

图 6 输入信号SNR与输出信号SNR的关系

Fig. 6. Input signal SNR vs. output signal SNR.

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

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 f_lmnp/f_LP01/arb.units

4束光都为LP₀₁模 1.000
4束光都为LP_11a或LP_11b模 0.747
2束光为LP₀₁模、2束光为LP_11a或LP_11b模 0.496
2束光为LP_11a模、2束光为LP_11b模 0.249

下载: 导出CSV

map

[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 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 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 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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空频复用光纤中四波混频过程的解析分析方法