 Solution

A high-numerical aperture objective focuses a laser to a steep intensity gradient. For a transparent particle with higher refractive index than its surroundings, refraction transfers photon momentum to the particle, giving a optical gradient force towards the intensity maximum. A forward scattering force pushes along the beam; a stable single-beam optical trap requires the restoring axial optical gradient force to exceed that push. In the small-particle limit, the same result follows from the induced-dipole potential energy, proportional to minus the local squared electric field. Close to the trap centre,An optical tweezers experiment can attach a bead to one end of a DNA molecule, hold the other end fixed, and obtain its force-extension relation from the bead displacement and calibrated trap stiffness. The measurement must distinguish the restoring force from Brownian motion and account for heating or optical damage.