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In the past few decades, although coupled-mode theory (CMT) has been extensively studied in quantum system, atomic system, plasmon system, circuit system, and so on, the theoretical origin is still plaguing many researchers. In the book of waves and fields in optoelectronics, the second-order differential equations of the simplest LC simple harmonic vibration circuit was turned into the first-order differential equation using the method of variable substitution by Haus. However, there is not loss in the simplest LC simple harmonic vibration circuit, loss term is introduced by qualitative analysis. Although this method of dealing with problems has no problems from a physical point of view, it is not rigorous enough from a mathematical point of view. In this paper, based on the secular perturbation theory, the well-known spring oscillator model is degenerated into two-mode CMT. Starting from the second-order differential equations of the spring oscillator model, the secular perturbation theory is used to obtain first order differential equations of two-mode CMT. The results show the relationships between each term’s coefficients in two-mode CMT and the physical quantities in Classical Mechanics are established by using the secular perturbation theory. Through solving two-mode coupled-mode equations, the energy transfer efficiency has been obtained. To verify the correctness of two-mode CMT, we design a coupled tuning fork mechanical vibration system, which consists of two experimental instruments to provide driving force and receive signals, two tuning forks and springs. The amplitude spectra are measured by an experimental instrument of forced vibration and resonance (HZDH4615), which provides a periodic driving signal for the tuning fork. To clarify the mechanism of the spectra, the numerical fitting has been performed by mathematica software based on the energy transfer efficiency. Theoretically, the obtained fitting parameters can also evaluate some important attributes of the system. The theoretical results are in close correspondence with the experiment. That is to say, two-mode CMT is suitable for classical vibration system.This study provides a more rigorous derivation for each term’s origin in two-mode CMT, and has guiding significance in the theoretical research of linear coupled vibration system.
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Keywords:
- linear coupled system /
- spring oscillator model /
- coupled-mode theory /
- secular perturbation theory
[1] Garrido A C L, Martinez M A G, Nussenzveig P 2002 Am. J. Phys. 70 37
[2] 朱旭鹏, 张轼, 石惠民, 陈智全, 全军, 薛书文, 张军, 段辉高 2019 物理学报 68 247301
Google Scholar
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Google Scholar
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Zhu X P, Shi H M, Zhang S, Chen Z Q, Zheng M J, Wang Y S, Xue S W, Zhang J, Duan H G 2019 Acta Phys. Sin. 68 247304
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Google Scholar
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Google Scholar
[6] LiuZV, Li J, Liu Z, Li W, Li J, Gu C, L iZ 2017 Sci. Rep. 7 8010
Google Scholar
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Google Scholar
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Google Scholar
[9] Marcuse D 1971 The Bell System Technical Journal 50 1791
Google Scholar
[10] Snyder A W 1970 IEEE Trans. Microwave Theory Tech. 18 383
Google Scholar
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Google Scholar
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Google Scholar
[14] Feng L, Xu YL, William S F, Lu M H, José E B O, Vilson R A, Chen Y F, Axel S 2012 Nat. Mater. 12 108
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 双模耦合体系的相关参数示意图 (a) 弹簧振子模型; (b) CMT模型
Figure 1. Parameters’ sketch of two-mode coupled system: (a) Spring oscillator model; (b) CMT model.
图 2 (a)两音叉耦合实验装置图; (b) 实验系统工作原理的示意图; (c)音叉单独振动时的实验谱和拟合谱; (d) 双音叉耦合下的实验谱和拟合谱
Figure 2. (a) Experimental device diagram of our two tuning forks’ coupled system; (b) schematic diagram of the experiment system; (c) measured and fitted spectra of two tuning forks without coupling; (d) measured and fitted spectra of two tuning forks with coupling.
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[1] Garrido A C L, Martinez M A G, Nussenzveig P 2002 Am. J. Phys. 70 37
[2] 朱旭鹏, 张轼, 石惠民, 陈智全, 全军, 薛书文, 张军, 段辉高 2019 物理学报 68 247301
Google Scholar
Zhu X P, Zhang S, Shi H M, Chen Z Q, Quan J, Xue S W, Zhang J, Duan H G 2019 Acta Phys. Sin. 68 247301
Google Scholar
[3] 朱旭鹏, 石惠民, 张轼, 陈智全, 郑梦洁, 王雅思, 薛书文, 张军, 段辉高 2019 物理学报 68 247304
Zhu X P, Shi H M, Zhang S, Chen Z Q, Zheng M J, Wang Y S, Xue S W, Zhang J, Duan H G 2019 Acta Phys. Sin. 68 247304
[4] Liu N, Langguth L, Weiss T, Kästel J, Fleischhauer M, Pfau T, Giessen H 2009 Nat. Mater. 8 758
Google Scholar
[5] Heylman K D, Thakkar N, Horak E H, Quillin S C, Goldsmith R H 2016 Nat. Photonics 10 788
Google Scholar
[6] LiuZV, Li J, Liu Z, Li W, Li J, Gu C, L iZ 2017 Sci. Rep. 7 8010
Google Scholar
[7] Suh W, Wang Z, Fan S 2004 IEEE J. Quantum Electron. 40 1511
Google Scholar
[8] Fan S, Suh W, Joannopoulos J D 2003 J. Opt. Soc. Am. A 20 569
Google Scholar
[9] Marcuse D 1971 The Bell System Technical Journal 50 1791
Google Scholar
[10] Snyder A W 1970 IEEE Trans. Microwave Theory Tech. 18 383
Google Scholar
[11] Hardy A, Streifer W 1985 J. Lightwave Technol. LT-3 1135
[12] McIntyre P D, Snyder A W 1973 J. Opt. Soc. Am. 63 1518
Google Scholar
[13] Butler J K, Ackley D E, Botez D 1984 Appl. Phys. Lett. 44 293
Google Scholar
[14] Feng L, Xu YL, William S F, Lu M H, José E B O, Vilson R A, Chen Y F, Axel S 2012 Nat. Mater. 12 108
[15] Hossein H, Absar U H, Steffen W, Hipolito GG, Ramy EG, Demetrios N C, Mercedeh K 2017 Nature 548 187
Google Scholar
[16] Mikhail F L, Mikhail V R, Alexander N P, Yuri S K 2017 Nat. Photonics 11 543
Google Scholar
[17] Miri M A, Andrea A 2019 Science 363 42
[18] SafaviNaeini A H, Alegre T P M, Chan J, Eichenfield M, Winger M, Lin Q, Hill J T, Chang D E, Painter O 2011 Nature 472 69
Google Scholar
[19] Andersen D R, Datta S, Gunshor R L 1983 J. Appl. Phys. 54 5608
Google Scholar
[20] Stegeman G I, Seaton C T 1985 J. Appl. Phys. 58 R57
Google Scholar
[21] Boller K J, Imamoğlu A, Harris S E 1991 Phys. Rev. Lett. 66 2593
Google Scholar
[22] Zhang S, Genov D A, Wang Y, Liu M, Zhang X 2008 Phys. Rev. Lett. 101 047401
Google Scholar
[23] Kurs A, Karalis A, Moffatt R, Joannopoulos J D, Fisher P, Soljačić M 2007 Science 317 83
Google Scholar
[24] Zhang F, Hackworth S A, Fu W, Li C, Mao Z, Sun M 2011 IEEE Trans. Magn. 47 1478
Google Scholar
[25] Karalis A, Joannopoulos J D, Soljačić M 2008 Ann. Phys. 323 34
Google Scholar
[26] Wang B, Yerazunis W, Teo K H 2013 Proc. IEEE 101 1359
Google Scholar
[27] Haus H A 1984 Waves and Fields in Optoelectronics (New Jersey: Prentice-Hall) pp197–217
[28] Huang W X, Wang Q J, Yin X G, Huang C P, Huang H, Wang Y M, Zhu Y Y 2011 J. Appl. Phys. 109 114310
Google Scholar
[29] Karabalin R, Cross M, Roukes M 2009 Phys. Rev. B 79 165309
Google Scholar
[30] Villanueva L, Kenig E, Karabalin R, Matheny M, Lifshitz R, Cross M, Roukes M 2013 Phys. Rev.Lett. 110 177208
Google Scholar
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