## 2 Materials and Methods  
The principle of the FLDI system depends on a phase difference owing to the nonuniform density generated by two orthogonal polarization beams passing through the flow field and obtaining a phase difference owing to the nonuniform density of the two optical paths. Here, the optical paths of the two beams, except for the rhomboid region in the middle, can be considered common path due to the tiny separation of the two beams, which indicates that the phase difference \(\Delta\varphi\) is mainly caused in the rhomboid region, especially the two convergence points. Then, the polarization state of the combined beam through a Wollaston prism is dependent on \(\Delta\varphi\), and the intensity I of the combined beam through the polarizer can be described as follows [29]  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos(\Delta\varphi)\]  
where \(I_{1}\) and \(I_{2}\) are the intensity of the two beams. To obtain the maximum interference contrast ratio, \(I_{1}\) and \(I_{2}\) should be identical, which can be realized by adjusting the half-wave plate. According to the above analysis, the time-varying density fluctuation is characterized by collecting the intensity variation, and the power spectrum of density fluctuation can be obtained by fast Fourier transform (FFT) processing. The separation distance of two orthogonally polarization beams on the focus lens changes as the Wollaston prism is in a different position. In other words, the distances of the virtual point sources introduced by the two divergent orthogonal polarization beams and the two convergence points within the measurement area change by horizontally adjusting the position of the Wollaston prism, which leads to the spatial resolution variation of the FLDI system. Therefore, the spatial resolution of the FLDI system can be adjusted by synchronously moving the Wollaston prisms. Figure 1 provides the schematic and image of the experimental setup of the FLDI system, which consists of the modules of the beam emitting portion and the beam receiving portion. In the beam emitting module, the laser source is a 532 nm continuous narrow linewidth pigtail-fiber laser delivered 20 mW power. The incident laser passes through a combination of a polarizer and half-wave plate, generating a linearly polarized beam with a high polarization ratio (over 20 dB) and polarization direction adjustment. A special pinhole is used to regulate the size of the beam to match up with the aperture of the Wollaston prism (W1). Two orthogonally polarization beams with a small divergence angle can be obtained through W1 with a divergence angle of less than 2\({}^{\prime}\). The two beams pass through the convex lens L1 and focus on two points with the separation distance \(\Delta x\) in the measurement area. Here, note that the separation distance \(\Delta x\) is defined as the spatial  
resolution in spanwise directions. Similar to the beam emitting module, the beam receiving module consists of a convex lens (L2) and a Wollaston prism (W2). The two orthogonal beams after the flow field are collected by utilizing L2 and merged into a single beam with a special polarization state through W2. A polarizer is used to convert polarization into intensity modulation, then the intensity of the beam is collected by a fast photodetector (D, DH-GDT-D020V) with a 10 MHz bandwidth. Here, the data acquisition system has a sampling rate of 1 MHz. In our experiment, the focal lengths of L1 and L2 are 250 mm. The convergence points are about 500 mm from the lenses L1 and L2. As mentioned before, the spatial resolution of this FLDI system can be adjusted by moving W1 and W2 synchronously which are installed on the two-dimensional moving stages, respectively. Figure 2a shows the separation distance of the two convergence points with different distances (50 mm-400 mm) between the Wollaston prism and the convex lens.