## 1 Introduction  
For the characterization and study of high-speed flow and combustion phenomena, non-intrusive measurement techniques based on optics and laser technology have attracted considerable attention in recent years. It has the potential to break through the limitations of traditional methods, such as thermocouple [1, 2] and pitot probes [3, 4], owing to the advantages of high precision, visualizing, non-interfering, wide range of measurements. Various non-intrusive measurement techniques, such as planar laser-induced fluorescence (PLIF) [5, 6, 7], resonantly focused schlieren deflectometry (REFSD) [8], nanoparticle-based planar laser scattering (NPLS) [9], and focused laser differential interferometry (FLD) [10, 11, 12, 13, 14, 15], have been developed to measure the density distribution and fluctuation of the flow field which are of great significance to research the boundary layer transition. Specifically, the FLDI system is a non-imaging, common path, polarizing interferometer, which converts the information of density fluctuation into light intensity modulation based on the Gladstone-Dale relation and interference. Owing to the rapid development of modern lasers and high-speed data-acquisition technologies, the FLDI system has the advantage of high spatial resolutions, high response frequency, and high flexibility, which can realize the measurement of density fluctuations at any interesting point without disturbance from other locations.  
The FLDI system was firstly proposed by [PERSON] et al., to reduce the line-of-sight integrating effect and measure the density fluctuation of wind tunnel and pedestal jet platforms [16, 17, 18, 19]. With the availability of high-speed data acquisition systems, the FLDI technique was developed rapidly in the last decade. In 2013, [PERSON] et al., used FLDI to study boundary layer instabilities for a cone in hypervelocity conditions and demonstrated reproducible measurements of the acoustic instability associated with the second Mack mode [20]. Recently, Huazhong University of Science and Technology studied the nonlinear behavior of boundary layer instability waves upon a flared-cone model using FLDI [21]. [PERSON] et al., used an FLDI system to investigate the freestream disturbances in the University of Southern Queensland's Mach 6 hypersonic wind tunnel [22]. [PERSON] et al., developed the quantitative measurement of the refractive index field by using a Mach-Zehnderinterferometer, and the response sensitivity of the FLDI system was explored by building a theoretical model of FLDI [23]. [PERSON] et al., analyzed the sensitivity function and effective measurement region of the FLDI system experimentally and theoretically [24]. Beyond density fluctuations, FLDI has been used for measurements of the streamwise convective velocity of disturbances, an analysis based on signal correlation between a pair of neighboring FLDI beams [25; 26; 27]. [PERSON] et al., measured the thickness of the wall boundary layer and the velocity profiles in the boundary layer with the single-point and two-point FLDI systems [28]. Furthermore, the accuracy of the measurement results is close to that of the traditional measurements. However, all of those reported works mentioned above are based on the fixed spatial resolution FLDI system, and the spatial resolutions are greater than 200 \(\upmu\)m, which depends heavily on the manufacturing process of birefringence crystals.  
This paper proposes an FLDI system with a tunable spatial resolution that covers a range of 83-382 \(\upmu\)m. Utilizing the spatial-resolution-tunable FLDI system, the density fluctuation characteristics of eight locations in a supersonic shear flow are measured. The results compare well to the numerical simulation data, which yield that the density fluctuation frequency decreases as the shear flow transmits, accompanied by the growth of the larger structures in the flow field.