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提出了一种基于测量反射率谱、使用Kramers-Kronig（KK）关系建立金属太赫兹色散模型的方法.结合合金铝和合金铜4-40 THz的测量反射率谱，通过反射系数振幅和相位的KK关系，采用高频端指数外推，低频端常数外推的方法，反演金属复折射率.以KK反演的复折射率作为实验值，以拟合复折射率和实验值误差最小为准则，使用遗传优化算法，拟合了合金铝和合金铜的Drude色散参数（等离子频率和碰撞频率）.基于优化的Drude模型计算了0.1-40 THz材料的复折射率，与椭偏仪的实测结果符合，验证了模型的准确性.该方法理论与实验相互验证，以测量的复折射率作为实验定标，将远红外频段的色散信息拓展到太赫兹频域，确定了太赫兹频段金属的微观物理参数，提供了太赫兹频段色散和散射机理的研究依据.
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关键词:
- 太赫兹 /
- 远红外 /
- Drude模型 /
- Kramers-Kronig关系
The extraction of terahertz dispersion parameters is confined in a limited region due to the limitation of the existing THz techniques. A method of studying the dispersion model of metals from the measurements of reflection spectrum and analysis of Kramers-Kronig (KK) relation is proposed. The reflection spectrum is measured by Vertex 80V Fourier transform spectrometer. In order to eliminate the signal noise of measured reflection spectrum, the measured spectrum is smoothed by Drude estimation. Using the smoothed reflection spectra of copper (Cu) alloy and aluminum (Al) alloy in a range of 440 THz, the complex refractivities are inversed based on the KK relation of amplitude and phase of reflective coefficient. The constant extrapolations at lower frequencies and the exponential extrapolation at higher frequencies are adopted in the KK integration. The exponential extrapolation index is adjusted according to the calibrating complex refractivity measured from far-infrared ellipsometer. According to the inversed complex refractivity, the plasma frequency and damping frequency in Drude model are optimized using the genetic algorithm. The objective function is defined as the error between the fitted complex refractivity and KK inversion. Since the optimal plasma frequency and damping frequency are different for different fitting frequencies, the obtained Drude parameters are averaged in order to reduce the influences of errors from KK inversion, measured reflection spectrum and calibrations. The complex refractivity indexes in a range from 15 THz to 40 THz, calculated by the established Drude model, are in good agreement with the measured calibrations from ellipsometer, which demonstrates the accuracy of the established Drude dispersion model. The reflection spectra below 4 THz are greatly distorted due to the signal noise, and the calibrating refractivity is located in the far infrared region, thus the complex refractivity is inversed in a region of 440 THz by KK algorithm. The complex refractivity indexes in a range of 0.120 THz, obtained by the proposed scheme, are for the vacancy, which will provide great support for the dispersion analysis in the whole terahertz gap. The procedures are helpful for extrapolating the dispersion information to terahertz band from the far infrared region. The scheme takes the advantage of the spectrometer and ellipsometer, and it requires high experimental precisions of reflection spectrum and calibrating refractivity. In addition, the scheme is adaptive to both metals and nonmetals by applying proper dispersion model which depends on the property of the reflection spectrum. The established model determines the microscopic dispersion parameters of material, which provides great support for the investigation of terahertz dispersion analysis, scattering mechanisms and imaging processes.-
Keywords:
- terahertz /
- far infrared /
- Drude model /
- Kramers-Kronig relation
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[2] Piesiewicz R, Jansen C, Mittleman D, Kleine-Ostmann T, Koch M, Krner T 2007 IEEE Trans. Antenn. Propag. 55 3002
[3] Chen Q 2012 M. S. Thesis (Harbin: Harbin Institute of Technology) (in Chinese) [陈琦 2012 硕士学位论文 (哈尔滨: 哈尔滨工业大学)]
[4] Wang L, Zhou Q 2007 Coll. Phys. 26 48 (in Chinese) [王磊, 周庆 2007 大学物理 26 48]
[5] Su J, Sun C, Wang X Q 2013 Optron. Lasers 24 408 (in Chinese) [苏杰, 孙诚, 王晓秋 2013 光电子激光 24 408]
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[8] Loewenstein E V, Smith D R, Morgan R L 1973 Appl. Opt. 12 398
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[10] Spitzer W G, Miller R C, Kleinman D A, Howarth L E 1962 Phys. Rev. 126 1710
[11] Spitzer W G, Kleinman D A 1961 Phys. Rev. 121 1324
[12] Hass M, Henvis B W 1962 J. Phys. Chem. Solids 23 1099
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[17] Silfsten P, Kontturi V, Ervasti T, Ketolainen J, Peiponen K E 2011 Opt. Lett. 36 778
[18] Duvillaret L, Garet F, Coutaz J L 1996 IEEE J. Sel. Top. Quantum Electron 2 739
[19] Yamashita T, Suga M, Okada T, Irisawa A, Imamura M 2015 40th International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz) Hong Kong, China, August 23-28, 2015 p1
[20] Kirley M P, Booske J H 2015 IEEE Trans. THz Sci. Technol. 5 1012
[21] Mou Y, Wu Z S, Gao Y Q, Yang Z Q, Yang Q J 2017 Infrared Phys. Technol. 80 58
[22] Cheng X H, Tang L G, Chen Z T, Gong M, Yu T J, Zhang G Y, Shi R Y 2008 Acta Phys. Sin. 57 5875 (in Chinese) [程兴华, 唐龙谷, 陈志涛, 龚敏, 于彤军, 张国义, 石瑞英 2008 物理学报 57 5875]
[23] Lucarini V, Peiponen K E, Saarinen J J, Vartiainen E M 2005 Kramers-Kronig Relations in Optical Materials Research (New York: Springer Berlin Heidelberg) pp27-50
[24] Wang R J, Deng B, Wang H Q, Qin Y L 2014 Acta Phys. Sin. 63 134102 (in Chinese) [王瑞君, 邓斌, 王宏强, 秦玉亮 2014 物理学报 63 134102]
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[1] Li Z, Cui T J, Zhong X J, Tao Y B, Lin H 2009 IEEE Antenn. Propag. Mag. 51 39
[2] Piesiewicz R, Jansen C, Mittleman D, Kleine-Ostmann T, Koch M, Krner T 2007 IEEE Trans. Antenn. Propag. 55 3002
[3] Chen Q 2012 M. S. Thesis (Harbin: Harbin Institute of Technology) (in Chinese) [陈琦 2012 硕士学位论文 (哈尔滨: 哈尔滨工业大学)]
[4] Wang L, Zhou Q 2007 Coll. Phys. 26 48 (in Chinese) [王磊, 周庆 2007 大学物理 26 48]
[5] Su J, Sun C, Wang X Q 2013 Optron. Lasers 24 408 (in Chinese) [苏杰, 孙诚, 王晓秋 2013 光电子激光 24 408]
[6] Ordal M A, Bell R J, Alexander Jr R W, Long L L, Querry M R 1985 Appl. Opt. 24 4493
[7] Ordal M A, Long L L, Bell R J, Bell S E, Bell R R, Alexander Jr R W, Ward C A 1983 Appl. Opt. 22 1099
[8] Loewenstein E V, Smith D R, Morgan R L 1973 Appl. Opt. 12 398
[9] Ohba T, Ikawa S I 1988 J. Appl. Phys. 64 4141
[10] Spitzer W G, Miller R C, Kleinman D A, Howarth L E 1962 Phys. Rev. 126 1710
[11] Spitzer W G, Kleinman D A 1961 Phys. Rev. 121 1324
[12] Hass M, Henvis B W 1962 J. Phys. Chem. Solids 23 1099
[13] Wills K, Knezevic I, Hagness S C 2013 Radio Science Meeting (Joint with AP-S Symposium) USNC-URSI Orlando, USA, July 7-13, 2013 p154
[14] Willis K J, Hagness S C, Knezevic I 2011 J. Appl. Phys. 110 063714
[15] Lucyszyn S 2004 IEEE Proc. -Microw. Antenn. Propag. 151 321
[16] Ordal M A, Bell R J, Alexander Jr R W, Newquist L A, Querry M R 1988 Appl. Opt. 27 1203
[17] Silfsten P, Kontturi V, Ervasti T, Ketolainen J, Peiponen K E 2011 Opt. Lett. 36 778
[18] Duvillaret L, Garet F, Coutaz J L 1996 IEEE J. Sel. Top. Quantum Electron 2 739
[19] Yamashita T, Suga M, Okada T, Irisawa A, Imamura M 2015 40th International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz) Hong Kong, China, August 23-28, 2015 p1
[20] Kirley M P, Booske J H 2015 IEEE Trans. THz Sci. Technol. 5 1012
[21] Mou Y, Wu Z S, Gao Y Q, Yang Z Q, Yang Q J 2017 Infrared Phys. Technol. 80 58
[22] Cheng X H, Tang L G, Chen Z T, Gong M, Yu T J, Zhang G Y, Shi R Y 2008 Acta Phys. Sin. 57 5875 (in Chinese) [程兴华, 唐龙谷, 陈志涛, 龚敏, 于彤军, 张国义, 石瑞英 2008 物理学报 57 5875]
[23] Lucarini V, Peiponen K E, Saarinen J J, Vartiainen E M 2005 Kramers-Kronig Relations in Optical Materials Research (New York: Springer Berlin Heidelberg) pp27-50
[24] Wang R J, Deng B, Wang H Q, Qin Y L 2014 Acta Phys. Sin. 63 134102 (in Chinese) [王瑞君, 邓斌, 王宏强, 秦玉亮 2014 物理学报 63 134102]
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