Analysis of Lorentzian Line Shape Function Broadened by Non-Sinusoidal Wavelength Modulation
SUN You-qun, WANG Yun-tao*, RUAN Chi, XU Song-song
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China
Abstract：In the present work, the Fourier analysis of Lorentzian line shape broadened by non-sinusoidal wavelength modulation was investigated, in which the third order and above harmonic items were ignored. The analytical expression of n-order Fourier coefficient was brought out, where a variable K named harmonic distortion to characterize the ratio of the second harmonic to the first harmonic was introduced. Numerical simulations based on the cases of K>0.01 and K<0.01 were carried out, and the result shows: non-sinusoidal modulation has little effect compared with the sinusoidal modulation when K value is less than 0.01, however, if K value is about 0.1 or higher, the center of the Fourier amplitude curve would deviate from the origin of coordinates. With the increase in the harmonic distortion, the deviation of the curve grows, and high order harmonics are more sensitive to the non-sinusoidal modulation compared with the low order harmonics. In addition, when harmonic distortion cannot be ignored, for example K>0.01, the effect of different depths of modulation on the odd and even order harmonic amplitude curve is significant. And the numerical simulation shows there exists an optimum value of modulation depth which could minimize the impact of the harmonic distortion, and both large K value and small K value would cause a great error. The conclusion of this work could be applied in error analysis of wavelength modulation spectroscopy system. And the results are helpful to deepening understanding of WMS and would be the important reference for some kind of frequency stabilization technology in laser instrument.
Key words：Wavelength modulation spectroscopy (WMS);Sinusoidal modulation;Second harmonic;Harmonic distortion;Lorentzian line shape
[1] Chao X, Jeffries J B, Hanson R K. Applied Physics B, 2012, 106(4): 987. [2] Gao G Z, Chen B X, Hu B. Measurement, 2013, 46(5): 1657. [3] Chao X, Jeffries J B, Hanson R K. Measurement Science and Technology, 2009, 20(11): 115201. [4] Hershberger W D. Journal of Applied Physics, 1948, 19(4): 411. [5] Arndt R. Journal of Applied Physics, 1965, 36: 2522. [6] Reid J, Labrie D. Applied Physics B, 1981, 26(3): 203. [7] Schmidt F M, Foltynowicz A, Gustafsson J, et al. Journal of Quantitative Spectroscopy and Radiative Transfer, 2005, 94(2): 225. [8] LU Jing, CAO Jia-nian, WANG Chun, et al(芦 婧, 曹家年, 王 淳, 等). Techniques of Automation and Applications(仪器仪表与检测技术), 2009, 28(3): 84. [9] Lins B, Zinn P, Engelbrecht R, et al. Applied Physics B, 2010, 100(2): 367. [10] Westberg J, Kluczynski P, Lundqvist S, et al. Journal of Quantitative Spectroscopy and Radiative Transfer, 2011, 112(9): 1443. [11] SUN Yan-guang, DONG Zuo-ren, CHEN Di-jun, et al(孙延光, 董作人, 陈迪俊, 等). Chinese Journal of Lasers(中国激光), 2013, 40(4): 0408002.
