Principles of Force-Measuring Optical Tweezers

A light beam carries momentum flux and when it is absorbed it creates a force F = dP/dt = nW/c where W is the power of the light beam (watts) and c/n is the speed of light in the surrounding medium (n=1.33 in water). When a light beam is refracted by an object, its momentum changes with the change in beam direction, thus imparting an equal but opposite reaction force to the refracting object.

Effect of external forces on light patterns leaving a trap: (A) Zero force:  laser beam enters through objective lens (OBJ) and traps bead at focus.  Beam then passes out the back side, is collimated by 2nd objective and projected onto detector surface.  Dotted lines indicate maximum collection angle (NA) of lenses.  (B) Force from above (red arrow) pushes light beam downward. Pattern on detector is offset down. (C) Same force applied from below  (D) External axial force pushing bead toward laser source.  Light pattern expands on detector.  (E) Axial force pushing bead away from laser source.  Light spot contracts and concentrates on detector.

 A light photon carries linear momentum P=h/l in the direction of its motion.  A ray of light carries linear momentum flux with magnitude |dP/dt|=nW/c where n is the refractive index of surrounding medium (e.g. water), W is the power of the ray in watts and c is the speed of light.  A transparent particle of high refractive index becomes trapped at a laser focus by virtue of the way it refracts the photons and changes their momenta.  If the trapped particle is somewhat offset from its equilibrium trap position (at the light focus) by an external force F, then the flux of light exiting the back side of the trap will undergo a change equal to the force, SdP/dt= F , where the summation is over all rays entering and leaving the trap.  Here a ray with no deflection cancels its contribution when entering and then leaving the focus.  By noting changes in the angular power distribution of the rays coming out the back side of the trap, it is possible to infer any external force on a trapped object, and hence on the light beam itself.  The figure above depicts the influence of external forces on the light leaving an optical trap.  Note how the exit light changes its ray pattern when forces are applied to the trap.  In practice, an optical trap formed by an underfilled objective lens (figure above) makes a very weak trap in the Z-axis direction. Therefore we normally use a counter-propagating beam geometry to form a symetric trap with 2 lasers focused to the same point (see figure below). Here the output light beams are projected by relay lenses and split by a non-polarizing beam splitter cubes and land on two separate detectors that measure the transverse forces (Fx and Fy) and the axial force (Fz) separately.