  Weak measurement has been proven to amplify the detection of changes in
meters while discarding most photons due to the low probability of
post-selection. Previous power-recycling schemes enable the failed
post-selection photons to be repeatedly selected, thus overcoming the
inefficient post-selection and increasing the precision of detection. In this
study, we focus on the polarization-based weak value angular-velocity
measurement and introduce three cyclic methods to enhance the accuracy of
detecting time shift in a Gaussian beam: power recycling, signal recycling, and
dual recycling schemes. By incorporating one or two partially transmitting
mirrors into the system, both the power and signal-to-noise ratio (SNR) of the
detected light are substantially enhanced. Compared to non-polarization
schemes, polarization-based approaches offer several advantages, including
lower optical loss, unique cyclic directions, and a wider optimal region. These
features effectively reduce crosstalk among different light paths and
theoretically eliminate the walk-off effect, thus yielding improvements in both
theoretical performance and application.
