A propagation of interferogram signal-to-noise (SNR) and phase uncertainty in Doppler asymmetric spatial heterodyne spectrometer
Authors: Sun Chen , Feng Yu-Tao , Fu Di , Zhang Ya-Fei , Li Juan , Liu Xue-Bin ,   Key Laboratory of Spectral Imaging Technology, Xi’an Institute of Optics and Precision Mechanics, CAS, Xi’an 710119, China,   University of Chinese Academy of Sciences, Beijing 100049, China
Published: December 2020

Passive atmospheric wind detection technique retrieves atmospheric wind profile by measuring the Doppler shift of airglow emissions. Doppler asymmetric spatial heterodyne spectrometer (DASH), which is a Fourier transform spectrometer(FTS), retrieves the Doppler shift information of airglow emissions by detecting the phase shift of interferograms, and the measured phase accuracy directly affects the retrieved wind speed precision. The signal-to-noise (SNR) ratio is one of the significant indexes for evaluating the performance of wind-measuring interferometers in engineering applications. Studying the quantitative relationship between retrieved phase uncertainty and original interferogram SNR that is based on observations is quite essential for the DASH design, performance evaluation and wind profile applications. In this paper, the study is based on the noise propagation theory in FTS and DASH phase retrieval model. According to the Fourier transform relationship between time and frequency domain, we start from original interferogram expression, then we conduct the Fourier transforming, single frequency extracting, inverse Fourier transforming, phase calculating and first-order Taylor expanding, and finally we establish a theoretical relationship model between original interferogram SNR and retrieved phase uncertainty. In order to verify the theoretical relationship model, firstly, we generate 20 groups of interferograms (each group with 1000 frames) randomly with varying the 30–250 times SNR value. After removing the low frequency baseline, we calculate the phase of each interferogram by DASH phase retrieval model, and obtain the phase uncertainty by calculating standard deviation of the 512th sampling of each group interferogram. Another phase retrieval uncertainty is obtained by using the theoretical relationship model between SNR and retrieved phase uncertainty derived from this paper. Secondly, a total of 23 groups of experimental interferograms (each group with 100 frames) with different intensities are collected through the self-developed DASH with a center wavelength of 632.8 nm, basic optical path difference of 50 mm, spectral resolution of 0.78 cm<sup>−1</sup>. Combining physical characteristics of shot noise and DASH parameters, interferogram SNR of each frame is calculated. We calculate phase uncertainty of experimental data through the two methods mentioned above. The results from the two different calculation methods are compared with each other to determine whether the conclusion is correct. In order to improve the accuracy of phase calculation, three lines are averaged as input to reduce the random error. The average residual between the two methods is only 0.03 mrad, the high consistency of the results indicates that the theoretical relationship model between SNR and retrieved phase uncertainty for DASH is correct. The phase uncertainty can be evaluated by interferogram SNR directly in engineering, which provides a theoretical basis for optimizing the interferometer design.