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We develop a novel hyperspectral imaging system using structured illumination in an SLM-based Michelson interferometer. In our design, we use a reflective SLM as a mirror in one of the arms of a Michelson interferometer, and scan the interferometer by varying the phase across the SLM display. For achieving the latter, we apply a checkerboard phase mask on the SLM display where the gray value varies between 0-255, thereby imparting a dynamic phase of up to 262° to the incident light beam. We couple a supercontinuum source into the interferometer in order to mimic an astronomical object such as the Sun, and choose a central wavelength of 637.4 nm akin to the strong emission line of Fe X present in the solar spectrum. We use a bandwidth of 30 nm, and extract fringes corresponding to a spectral resolution of 3.8 nm which is limited by the reflectivity of the SLM. We also demonstrate a maximum wavelength tunability of ~8 nm by varying the phase over the phase mask with a spectral sampling of around 0.03 nm between intermediate fringes. The checkerboard phase mask can be adapted close to real time on time-scales of a few tens of milliseconds to obtain spectral information for other near-contiguous wavelengths. The compactness, potential low cost, low power requirements, real-time tunability and lack of moving mechanical parts in the setup implies that it can have very useful applications in settings which require near real-time, multi-wavelength spectroscopic applications, and is especially relevant in space astronomy.