  High-brilliance high-polarization $\gamma$ rays based on Compton scattering
are of great significance in broad areas, such as nuclear, high-energy,
astro-physics, etc. However, the transfer mechanism of spin angular momentum in
the transition from linear, through weakly into strongly nonlinear processes is
still unclear, which severely limits the simultaneous control of brilliance and
polarization of high-energy $\gamma$ rays. In this work, we investigate the
manipulation mechanism of high-quality polarized $\gamma$ rays in Compton
scattering of the ultrarelativistic electron beam colliding with an intense
laser pulse. We find that the contradiction lies in the simultaneous
achievement of high-brilliance and high-polarization of $\gamma$ rays by
increasing laser intensity, since the polarization is predominately contributed
by the electron (laser photon) spin via multi-photon (single-photon) absorption
channel. Moreover, we confirm that the signature of $\gamma$-ray polarization
can be applied for observing the nonlinear effects (multi-photon absorption) of
Compton scattering with moderate-intensity laser facilities.
