  A relatively efficient scheme to generate energetic $\gamma$-ray is
all-optical Compton scattering based on wakefield acceleration and plasma
mirror where planar or concave solid target is usually employed to reflect
driving pulse. For flat plasma mirror case, the colliding pulse radius is far
larger than the radius of accelerated electron beam, which leads to smaller
laser utilization efficiency. The concave plasma mirror can efficiently focus
reflected laser to match electron beam radius. But the produced photon beam
collimation is worse due to intensive transverse field and larger transverse
wave vector of colliding laser. Here, we propose a new mechanism to achieve
Compton scattering via employing convex plasma mirror. The convex plasma mirror
improves reflected laser longitudinal length in addition to focusing effect,
which ensures longer colliding time and larger laser utilization efficiency.
Importantly, it leads transverse wave vector of colliding laser to be smaller,
which guarantees better photon bunch collimation.
