The paper covers the description of a fiber optic sensor of linear displacement with a single multimode fiber and a movement mirror positioned near the output end of the fiber perpendicularly to its axis. The value of the signal of the considered sensor is determined by the rate of a beam reflected from mirror and introduced back into the fiber. New methods of increase in the band of measured shifts and enhancement of the sensor sensitivity are described in the paper. In particular, we propose to fill a space between the fiber face and the mirror by the light-absorbing liquid and to use polarization splitting of input and output beams. Mathematical descriptions of the sensitive element and the polarization beam splitting scheme are considered in the paper. We used a method of geometrical optics in Hamiltonian formulation for mathematical sensor modeling. Presented experimental results confirm the theoretical conclusion.