用久期微扰理论将弹簧振子模型退化为耦合模理论

朱存远; 李朝刚; 方泉; 汪茂胜; 彭雪城; 黄万霞

doi:10.7498/aps.69.20191505

摘要

尽管耦合模理论在过去几十年内已经被广泛研究, 但它的理论来源还是困扰着广大研究者. 在这里, 基于久期微扰理论, 将经典弹簧振子模型退化为耦合模理论, 并将该理论用于解释音叉耦合的实验现象. 研究表明这种方法将耦合模理论中每一项的系数都与经典力学中的相关物理量建立关联, 且理论和实验结果符合得很好. 该研究为耦合模理论中每一项的来源提供了一种较严谨的推导方法, 在线性耦合体系的理论研究方面有一定的指导意义.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
安徽师范大学物理与电子信息学院, 芜湖　241002

2.
复旦大学, 应用表面物理国家重点实验室, 上海　200433

通信作者: 黄万霞, kate@mail.ahnu.edu.cn

基金项目: 国家级-国家自然科学基金面上项目(11304002)

Authors and contacts

1.
School of Physics and Electronic Information, Anhui Normal University, Wuhu 241002, China

2.
State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China

Corresponding author: Huang Wan-Xia, kate@mail.ahnu.edu.cn

文章全文 : translate this paragraph

参考文献

[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
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[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
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[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
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[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
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施引文献

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

Fig. 1. Parameters’ sketch of two-mode coupled system: (a) Spring oscillator model; (b) CMT model.

下载: 全尺寸图片幻灯片

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

Fig. 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.

下载: 全尺寸图片幻灯片

[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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用久期微扰理论将弹簧振子模型退化为耦合模理论