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计算电容中Fabry-Perot干涉仪测量位移的相位修正方法

王建波 钱进 刘忠有 陆祖良 黄璐 杨雁 殷聪 李同保

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计算电容中Fabry-Perot干涉仪测量位移的相位修正方法

王建波, 钱进, 刘忠有, 陆祖良, 黄璐, 杨雁, 殷聪, 李同保

Methode of phase correction of displacement measurement using Fabry-Perot interferometer in calculable capacitor

Wang Jian-Bo, Qian Jin, Liu Zhong-You, Lu Zu-Liang, Huang Lu, Yang Yan, Yin Cong, Li Tong-Bao
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  • 计算电容是复现电学阻抗单位的基准装置, 利用计算电容值和量子霍尔电阻值可以准确计算出精细结构常数. 计算电容的本质是通过高准确度地测量屏蔽电极的位移, 实现对电容量值的测量. 因此, 基于Fabry-Perot干涉仪的精密电极位移测量系统是计算电容装置中最为核心和关键的部分. 在Fabry-Perot干涉仪测位移过程中, 由于高斯激光束存在轴向Gouy相位, 该附加相位将会引起相邻干涉条纹对应位移的变化(大于或者小于/2), 导致位移的测量值与实际值存在偏差. 本文阐述了高斯激光场的传播特性, 利用高斯激光束在自由空间和透过薄透镜复振幅的变换关系, 建立了计算电容装置中Fabry-Perot干涉仪透射光束的传输模型; 通过对不同腔长的Fabry-Perot干涉仪透射光场相位的分析, 获得了高斯激光束轴向Gouy相位修正与传输距离的关系. 结果表明, 当腔长从111.3 mm移动至316.3 mm时, 在接收距离为560 mm的情况下, 高斯光束轴向Gouy 相位引起的位移修正的绝对值最小为0.7 nm, 其相对相位修正量|L|/|L| = 3.410-9.
    The calculable capacitor is a classical and fundamental experimental apparatus in precision electromagnetic measurements. It is the alternating current (AC) impedance primary standard, and an important tool in measuring the fine structure constant. The calculable capacitor provides a way to directly link the capacitance unit to the mechanical unit of length. In the calculable capacitor, the displacement measurement of the guard electrode is an essential part, because the average value of the cross capacitances is directly proportional to the linear displacement of the moving guard electrode. In order to measure the displacement with a high accuracy of 10-9 or lower, a Fabry-Perot interferometer, whose cavity length is traceable to a stabilized laser by the phase sensitive detection technique, is employed. Considering that the Fabry-Perot interferometer is irradiated by the Gaussian laser beam, the effect of the phase shift of the Gaussian field, relative to the plane wave, should be carefully considered in the displacement measurement. The amplitude of the Gaussian laser beam disperses out of the region where it can be assumed to be plane-wave propagation, so its wavefronts bend and their spacing is different from that of the plane wave. As a result, the corresponding distance of an interference fringe from the coherent Gaussian laser beams is not strictly equal to /2, and it means that the displacement correction based on the phase shift of the Gaussian laser beam in the Fabry-Perot interferometer is inevitable. Therefore, the measured result should add or subtract the correction value to obtain the actual displacement of the interferometer. In order to determine the Gouy phase correction, an interferometer model based on the calculable capacitor is studied analytically and numerically. Using the free space propagation and lens transformation of the Gaussian beam field, the complex amplitude of the partial beam transmitted through the interferometer is obtained, and its phase versus the longitude propagation distance is analyzed. The amplitude and phase of the total transmitted beam, which is the coherent superposition of all the partial beams, are presented. Since the Fabry-Perot interferometer in the calculable capacitor is actively locked to a stabilized laser at two different cavity lengths, the phase of the transmitted beam at each cavity length is calculated individually. The phase difference between the two transmitted beams versus the longitude propagation distance is also analyzed numerically. The simulation result demonstrates that the minimum value of the displacement correction can be obtained by actively detecting the laser light at a distance of 560 mm from output mirror, when the Fabry-Perot interferometer moves from the cavity length of 111.3 mm to 316.3 mm, and it means that a displacement correction value of 0.7 nm, with a relative value of |L|/|L| = 3.410-9, should be added to the measured displacement of the guard electrode.
      通信作者: 钱进, qianjin@nim.ac.cn
    • 基金项目: 国家重大科学仪器设备开发专项(批准号: 2012YQ10022503)和质检公益性行业科研专项(批准号: 20150002)资助的课题.
      Corresponding author: Qian Jin, qianjin@nim.ac.cn
    • Funds: Project supported by the National Key Scientific Instrument and Equipment Development Project (Grant No. 2012YQ10022503), and the Special Scientific Research Fund of Quality Inspection of Public Welfare Profession of China (Grant No. 20150002).
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    Thompson A M, Lampard D G 1956 Nature 177 888

    [2]

    Klitzing K v, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494

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    Mohr P J, Taylor B N, Newell D B 2012 Rev. Mod. Phys. 84 1527

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    Cutkosky R D 1961 J. Res. Nat. Bur. Stand. 65A 147

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    Clothier W K 1965 Metrologia 1 36

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    Thompson A M 1959 Proceedings of the IEE - Part C:Monographs 104 271

    [7]

    Bachmair H, Funck T, Hanke R, Lang H 1995 IEEE Trans. Instrum. Meas. 44 440

    [8]

    Igarashi T, Kanno M, Koizumi Y, Haneda K 1970 IEEE Trans. Instrum. Meas. 19 297

    [9]

    Jeffery A M, Elmquist R E, Lee L H, Shields J Q, Dziuba R F 1997 IEEE Trans. Instrum. Meas. 46 264

    [10]

    Jones K, Corney A C 1987 Metrologia 24 1

    [11]

    Small G W 1996 Conference on Precision Electromagnetic MeasurementsLaguna Beach, California, USA, June 10-12, 1999 p8

    [12]

    Cross Capacitor Group of National Institute of Metrology 1980 Acta Metrol. Sin 1 16 (in Chinese) [中国计量科学研究院计算电容组 1980 计量学报 1 16]

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    Zhang Z, Lu Z 1982 Acta Metrol. Sin. 3 250

    [14]

    Lu Z, Huang L, Yang Y, Zhao J, Qian J, Lu W, Liu Z, Zhang Z, Liu X, Wang J, Wang W, Lu Y, He Q 2015 IEEE Trans. Instrum. Meas. 64 1496

    [15]

    Shields J Q, Dziuba R F, Layer H P 1989 IEEE Trans. Instrum. Meas. 38 249

    [16]

    Lawall J R 2005 J. Opt. Soc. Am. A 22 2786

    [17]

    Fletcher N, Goebel R, Robertsson L, Stock M 2004 Conference on Precision Electromagnetic Measurements, London, England, June 27-July 2, 2004 p485

    [18]

    Andreas B, Ferroglio L, Fujii K, Kuramoto N, Mana G 2011 Metrologia 48 S104

    [19]

    Kogelnik H, Li T 1966 Appl. Opt. 5 1550

    [20]

    Boyd R W 1980 J. Opt. Soc. Am. 70 877

    [21]

    Feng S M, Winful H G 2001 Opt. Lett. 26 485

    [22]

    Martelli P, Tacca M, Gatto A, Moneta G, Martinelli M 2010 Opt. Exp. 18 7108

    [23]

    Tyc T 2012 Opt. Lett. 37 924

    [24]

    Wu X F, Deng D M, Guo Q 2011 Chin. Phys. B 20 84201

    [25]

    Zhou Y H, Jiang H B, Gong Q H 2006 Chin. Phys. Lett. 23 110

    [26]

    Lennart R 2007 Metrologia 44 35

    [27]

    Coddington I, Swann W C, Nenadovic L, Newbury N R 2009 Nature Photon. 3 351

    [28]

    Small G W, Fiander J R 2011 IEEE Trans. Instrum. Meas. 60 2489

    [29]

    Siegman A E 1986 Lasers (Palo Alto: University Science Books) pp637-667

    [30]

    Lu Z L, Huang L, Yang Y, Zhao J T, Qian J, Lu W J, Liu Z Y, Zhang Z H, Liu X Y, Wang J B, Wang W, He X B 2014 Acta Metrol. Sin 35 521 (in Chinese) [陆祖良, 黄璐, 杨雁, 赵建亭, 钱进, 陆文骏, 刘忠有, 张钟华, 刘秀英, 王建波, 王维, 何小兵 2014 计量学报 35 521]

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出版历程
  • 收稿日期:  2015-12-26
  • 修回日期:  2016-02-28
  • 刊出日期:  2016-06-05

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