  We present a weak measurement protocol that permits a sensitive estimation of
angular rotations based on the concept of weak-value amplification. The shift
in the state of a pointer, in both angular position and the conjugate orbital
angular momentum bases, is used to estimate angular rotations. This is done by
an amplification of both the real and imaginary parts of the weak-value of a
polarization operator that has been coupled to the pointer, which is a spatial
mode, via a spin-orbit coupling. Our experiment demonstrates the first
realization of weak-value amplification in the azimuthal degree of freedom. We
have achieved effective amplification factors as large as 100, providing a
sensitivity that is on par with more complicated methods that employ quantum
states of light or extremely large values of orbital angular momentum.
