A MEMS apparatus for scanning an optical beam comprises a mirror operative to perform a rotational motion to a maximum rotation angle around a mirror rotation axis, and a bouncing mechanism operative to provide a bouncing event and to reverse the rotational motion. The bouncing event provides the mirror with a piecewise linear response to actuation by intrinsically nonlinear electrostatic forces. The bouncing mechanism includes an element chosen to impart an overall nonlinear stiffness to the system and is selected from the group of elements consisting of a bouncer and a pre-curved nonlinear stiffness element.