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The mechanical properties of materials hold significant importance in fundamental research, material, mechanical and civil engineering. These properties are often characterized by some elastic parameters, e.g., Young's modulus, shear modulus and Poisson's ratio. Self-mixing interferometry (SMI) is a unique non-destructive sensing technology that requires minimal components. In this work, we have developed a time-multiplexed fiber-coupled laser SMI sensor in conjunction with impulse excitation technique to simultaneously measure Young modulus, shear modulus and Poisson's ratio of materials. Two specimens made of brass and aluminum 6061 were assessed to validate the feasibility of the designed sensor. The measured values of Young's modulus (101.2 GPa for brass and 69.5 GPa for aluminum 6061), shear modulus (36.3 GPa for brass and 29.5 GPa for aluminum 6061), and Poisson's ratio (0.39 for brass and 0.34 for aluminum 6061) aligned with values reported in existing l ....