## 2 Principle of DASH Interferometer  
### Principle of DASH  
The DASH interferometer, which is based on Spatial Heterodyne Spectroscopy [19], is an asymmetric spatial heterodyne interferometer. It allows a sufficiently high spectral resolution by introducing a large additional optical path offset to passively measure the small wavelength shifts of emission lines caused by Doppler shifts from speeds, thereby measuring the line-of-sight velocity. Figure 1 shows the schematic of the DASH interferometer technique. The DASH interferometer receives a telecentric incident beam. The incident wavefront is split at the beam splitter (BS), so that the two resulting beams travel along the arms of different path lengths. The two beams illuminate the gratings (G1 and G2) at the end of the interferometer arms. After diffraction at the gratings, the wavefronts return to the beam splitter and recombine. Because one wavefront is delayed by the optical path offset, both the wavefronts are tilted by the angle with respect to the optical axis at this point. The detector array records a Fizeau parallel fringe pattern. The interferogram can be written as follows [20]:  
\[I(x)=\frac{1}{2}\int_{0}^{\infty}B(\sigma)[1+\cos\{2\pi[4(\sigma-\sigma_{L}) \tan\theta_{L}]x+2\sigma\Delta d\}]d\sigma, \tag{1}\]  
where \(\sigma\) is the wave number of incident light, \(\sigma_{L}\) is the Littrow wavenumber, \(\theta_{L}\) is the Littrow angle, \(B(\sigma)\) is the incident spectral density, \(\Delta d\) is the path difference offset, and \(x\) is the location on the detector array (\(x\) = 0 is the center of the array). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Structure** & **Advantage** & **Limitation** \\ \hline \multirow{2}{*}{Mounting an FPI with six epoxy pads} & Micro-stress mounting with low heat & Mounting an interferometer with a rotationally symmetric structure \\ \hline Mesh bonding surface for mounting the & \multirow{2}{*}{Micro-stress mounting} & \multirow{2}{*}{Larger bonding area} \\ \multicolumn{2}{c}{Sagnac interferometer prism} \\ \hline Three loaded plungers and precision & \multirow{2}{*}{Thermally stable mounting} & \multirow{2}{*}{Used in lab instrument} \\ spacers for mounting split SHS & & \\ \hline Combination of four bonding surfaces for & Stable mounting by improving & Used in mounting a normal-sized \\ mounting DASH & natural frequency & DASH interferometer \\ \hline Side adhesive structure & Support large optical components & Used to support large transmission and \\  & & reflection prisms \\ \hline \multirow{4}{*}{Proposed flexible structure} & Micro-stress mounting structure with low & \multirow{2}{*}{-} \\ \multicolumn{2}{c}{heat conduction to support a} & & \\ \multicolumn{2}{c}{large-sized interferometer} & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structure for mounting the interferometer. Figure 1: Schematic of the DASH interferometer technique. When the Doppler shift is introduced by the Doppler velocity, the wave number of incident light is given:  
\[\sigma=\sigma_{0}+\Delta\sigma, \tag{2}\]  
where \(\Delta\sigma\) is the spectral resolution and \(\sigma_{0}\) is the wavenumber of the line center. Then, Equation (1) can be written as:  
\[I(x)=\frac{1}{2}\int_{0}^{\infty}B(\sigma)[1+\cos\{2\pi[4(\sigma_{0}+\Delta \sigma-\sigma_{L})\tan\theta_{L}]x+2(\sigma_{0}+\Delta\sigma)\Delta d\}]d\sigma, \tag{3}\]  
Using Fourier transform, extracting the characteristic peaks and inverse Fourier transform, the complex interferogram can be obtained:  
\[I_{D}(x)=\frac{1}{2}SE(x)[\cos\varphi(x)+i\sin\varphi(x)], \tag{4}\]  
where \(S\) is the intensity coefficient of the spectral line, \(E(x)\) is the spectral envelope function, and \(\varphi(x)\) is the interferogram phase. Then, the phase can be calculated from the ratio of its imaginary \(\Im\) and its real part \(\Im\)[21].