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本文推导了单模光纤中的声波亥姆霍兹方程, 利用分离变量法求解并获得正规声波导导模的特征方程, 定义了声模的归一化频率, 结合贝塞尔函数的宗量近似分析了声波模式的特征值范围、截止频率和远离截止, 探讨了声模的色散和布里渊增益谱的多峰成因. 研究结果表明单模光纤中纵向声波基模L01模无截止, 主要被限制在纤芯中, 与光基模耦合形成布里渊增益谱的主峰; 高阶声模都存在低频截止, 在包层分布比基模多, 与LP01模耦合形成布里渊增益谱的次峰. 只有纵向L0n声模对后向布里渊增益谱有贡献, 纤芯掺锗浓度增大能使布里渊增益谱发生红移, 声模数量增多, L01模的增益峰值逐渐变大而高阶模的贡献减小. 泵浦波长为1.55 μm, 纤芯掺锗浓度3.65%、纤芯半径4.2 μm的单模光纤存在4个L0n和16个Lmn (m > 0)声模, 声模L01, L03, L04与光模LP01声光耦合产生布里渊增益谱的1个主峰和2个弱峰; 纤芯掺锗浓度15%, 纤芯半径1.3 μm的单模光纤存在3个L0n模和7个Lmn (m > 0)模, L01, L02, L03模与LP01模声光耦合使得布里渊增益谱呈现3个主峰. 这些结论可以完全解释相应的实验现象, 也为光纤SBS声波导研究及应用提供理论参考.In this work, the acoustic Helmholtz equation is derived, and its analytical solution and the characteristic equation of the uniform guide mode in single mode fibers are obtained by the method of separation of variables. The normalized frequency of the acoustic mode is defined. By combining the argument approximation of the Bessel function are analyzed the eigenvalue range of the acoustic mode, the cut-off frequency, far from the cut-off state of the acoustic mode induced by backward stimulated Brillouin scattering, the dispersion and the multi-peak Brillouin gain spectrum. The research results indicate that the longitudinal acoustic fundamental mode L01 cannot be cut-off and is mainly confined in the fiber core, which is coupled with the optical fundamental mode LP01 to form the main peak of the Brillouin gain spectrum. The other higher-order acoustic modes all have low cut-off frequencies and are distributed more in the fiber cladding than mode L01 which couples with the optical fundamental mode LP01 to form the subpeaks of the Brillouin gain spectrum. The transverse normalized phase constant and effective refractive index of the acoustic mode increase with normalized frequency increasing. Only longitudinal acoustic modes L0n contribute to backward Brillouin gain spectrum in single mode fiber. When the GeO2 concentration is less than 4% and core radius is 4.5 μm, the single mode characteristics of the fiber remain unchanged, but the maximum number of acoustic L0n modes is 4. With the increase of GeO2 concentration in the fiber core, the Brillouin gain spectrum is red-shifted and the number of acoustic modes increases, the Brillouin gain peak value of L01 mode gradually increases, and the contributions of higher-order modes decrease. The single-mode fiber with a core’s germanium doped concentration of 3.65% and core radius of 4.3 μm has 4 L0n modes and 16 Lmn (m>0) modes at a wavelength of 1.55 μm, with one main peak and two subpeaks in the Brillouin gain spectrum appearing due to the acousto-optic coupling of the acoustic L01, L03, and L04 modes with the optical LP01 mode. The single-mode fiber with a core’s germanium doped concentration of 15% and core radius of 1.3 μm has 3 L0n modes and 7 Lmn (m>0) modes, with the Brillouin gain spectrum having 3 main peaks due to the acousto-optic coupling of the L01, L02, and L03 modes with the LP01 mode. These conclusions are well consistent with the reported experimental phenomena and provide theoretical references for studying and utilizing the SBS acoustic waveguide in optical fibers.
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图 1 (a) L0n模; (b) L1n模的Ua取值范围
Fig. 1. The Ua value range of (a) L0n mode and (b) L1n mode
图 2 声模色散曲线 (a) Ua与Va的关系; (b) na, eff与Va的关系
Fig. 2. Dispersion curve of acoustic mode: (a) $U_{{\mathrm{a}}}=f(V_{{\mathrm{a}}}) $; (b) $n_{{\mathrm{a,eff}}}=f(V_{\mathrm{a}}) $.
图 5 Fiber 1的(a) L0n模特征方程和(b)归一化场分布
Fig. 5. (a) L0n mode characteristic equation and (b) normalized field distribution of Fiber 1.
图 7 Fiber 2的(a) L0n模特征方程和(b)归一化场分布
Fig. 7. (a) L0n mode characteristic equation and (b) normalized field distribution of Fiber 2.
表 2 Fiber 1和Fiber 2的Lmn模BFS (GHz)
Table 2. BFS (GHz) of the Lmn modes in Fiber 1 and Fiber 2.
Fiber type m n 1 2 3 4 Fiber 1 0 10.9244 10.9694 11.0488 11.1576 1 10.9407 11.0032 11.0985 — 2 10.9620 11.0418 11.1576 — 3 10.9880 11.0848 11.2052 — 4 11.0185 11.1318 — — 5 11.0532 11.1812 — — 6 11.0921 — — — 7 11.1349 — — — 8 11.1814 — — — Fiber 2 0 10.0902 10.4721 11.0857 — 1 10.2306 10.7466 — — 2 10.4117 11.0438 — — 3 10.6287 — — — 4 10.8779 — — — 5 11.1511 — — — 
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表 3 两种单模光纤的理论计算与实验结果比较
Table 3. Comparison of experimental and theoretical calculation results of two single-mode optical fibers.
Fiber type Aeff/μm2 Aao/μm2 I BFS/GHz Relative error/% Reference This paper Fiber 1 76.32 78.87 0.9668 10.9170[22] 10.9244 0.0678 15612.35 0.0049 10.9630[22] 10.9694 0.0584 14173.60 0.0054 11.0430[22] 11.0488 0.0525 11249.48 0.0068 11.1540[22] 11.1576 0.0323 Fiber 2 21.02 26.06 0.8068 10.0000[23] 10.0902 0.8087 482.47 0.0436 10.5000[23] 10.4721 0.2657 328.89 0.0639 11.1100[23] 11.0857 0.2187
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[1] Shimizu K, Horiguchi T, Koyamada Y, Kurashima T 1993 Opt. Lett. 18 185
Google Scholar
[2] Bao X Y, Chen L 2012 Sensors-Basel 12 8601
Google Scholar
[3] Li M J, Xing C, Wang J, Gray S, Liu A P, Demeritt J A, Ruffin A B, Crowley A M, Walton D T, Zenteno L A 2007 Opt. Express 15 8290
Google Scholar
[4] Gonzalez H M, Song K Y, Thévenaz L 2005 Appl. Phys. Lett. 87 081113
Google Scholar
[5] Hon D T 1980 Opt. Lett. 5 516
Google Scholar
[6] Jen C K, Neron C, Shang A, Abe K, Bonnell L, Kushibiki J 1993 J. Am. Ceram. Soc. 76 712
Google Scholar
[7] Waldron R 1969 IEEE Trans. Microwave Theory Tech. 17 893
Google Scholar
[8] Shelby R, Levenson M, Bayer P 1985 Phys. Rev. B 31 5244
Google Scholar
[9] Shibata N, Okamoto K, Azuma Y 1989 J. Opt. Soc. Am. B 6 1167
Google Scholar
[10] Tartara L, Codemard C, Maran J N, Cherif R, Zghal M 2009 Opt. Commun. 282 2431
Google Scholar
[11] Poulton C G, Pant R, Eggleton B J 2013 J. Opt. Soc. Am. B 30 2657
Google Scholar
[12] Xing C, Ke C J, Guo Z, Yang K J, Wang H Y, Zhong Y B, Liu D M 2018 Opt. Express 26 28793
Google Scholar
[13] Tsvetkov S V, Likhachev M E 2023 B. Lebedev. Phys. Inst+. 50 291
Google Scholar
[14] Huang B, Wang J Q, Shao X P 2023 Photonics-Basel 10 282
Google Scholar
[15] He D Y, Liao M S, Hu L L, Yu C L, Qi Y F, Shen H, Chen L, Yang Q B, Liu M Z, Wang M, Zhou Q L, Gao W Q, Wang T X 2023 Opt. Express 31 1888
Google Scholar
[16] Kobyakov A, Kumar S, Chowdhury D Q, Ruffin A B, Sauer M, Bickham S R, Mishra R 2005 Opt. Express 13 5338
Google Scholar
[17] Dong Y, Ren G B, Xiao H, Gao Y X, Li H S, Xiao S Y, Jian S S 2017 Ieee Photonic Tech. L. 29 1955
Google Scholar
[18] 吴重庆 2000光波导理论(北京: 清华大学出版社) 第41—43页
Wu C Q 2000 Optical Waveguide Theory (Beijing: Tsinghua University Press) pp41–43
[19] Zou W W, He Z Y, Hotate K 2008 Opt. Express 16 10006
Google Scholar
[20] Jia D F, Ge C F 2004 Nonlinear Fiber Optics (Beijing: Electronic Industry Press) 第246—247页
贾东方, 葛春风 2014非线性光纤光学(北京: 电子工业出版社) pp246–247
[21] Ruffin A B, Li M J, Chen X, Kobyakov A, Annunziata F 2005 Opt. Lett. 30 3123
Google Scholar
[22] Zou W W, He Z Y, Hotate K 2006 Ieee Photonic Tech. L. 18 2487
Google Scholar
[23] Dasgupta S, Poletti F, Liu S, Petropoulos P, Richardson D J, Grüner-Nielsen L, Herstrøm S 2011 J. Lightwave Techno. 29 22
Google Scholar
[24] Koyamada Y, Sato S, Nakamura S, Sotobayashi H, Chujo W 2004 J. Lightwave Techno. 22 631
Google Scholar
[25] Kobyakov A, Sauer M, Chowdhury D 2010 Adv. Opt. Photonics 2 1
Google Scholar
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