  We report on the use of an interferometric weak value technique to amplify
very small transverse deflections of an optical beam. By entangling the beam's
transverse degrees of freedom with the which-path states of a Sagnac
interferometer, it is possible to realize an optical amplifier for polarization
independent deflections. The theory for the interferometric weak value
amplification method is presented along with the experimental results, which
are in good agreement. Of particular interest, we measured the angular
deflection of a mirror down to 560 femtoradians and the linear travel of a
piezo actuator down to 20 femtometers.
