Deflection measurements with weak values

We report on the use of an interferometric weak value technique to amplify small transverse deflections of an optical beam. Our techniques consists of coupling the beam's transverse degrees of freedom to the which-path states of a Sagnac interferometer. The theory for the amplification, including signal to noise ratio considerations, 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 showed the signal to noise ratio to be amplified by a factor of 150.