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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.
[1] Thompson A M, Lampard D G 1956 Nature 177 888
[2] Klitzing K v, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[3] Mohr P J, Taylor B N, Newell D B 2012 Rev. Mod. Phys. 84 1527
[4] Cutkosky R D 1961 J. Res. Nat. Bur. Stand. 65A 147
[5] Clothier W K 1965 Metrologia 1 36
[6] 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]
[13] 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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[1] Thompson A M, Lampard D G 1956 Nature 177 888
[2] Klitzing K v, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
[3] Mohr P J, Taylor B N, Newell D B 2012 Rev. Mod. Phys. 84 1527
[4] Cutkosky R D 1961 J. Res. Nat. Bur. Stand. 65A 147
[5] Clothier W K 1965 Metrologia 1 36
[6] 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]
[13] 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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