Improvement of all-optical Compton $\gamma$-rays source by reshaping colliding pulse

All-optical Compton scattering is a remarkable method of generating high-quality $\gamma$ radiation source. It is easier achieved in experiment by employing a pulse based on laser wakefield accelerator. The driving laser is backward reflected when wakefield acceleration stage is over and thus it naturally collides with energetic electrons. To increase reflected pulse intensity, parabolic focusing plasma mirror instead of flat reflecting target is usually adopted. But concave focusing mirror also deteriorates the emitted photon beam monochromaticity and collimation. We propose using stepped focusing plasma mirror to reflect the driving pulse to conquer these issues. The longitudinal length of reflected pulse by stepped target is larger and intensity is relatively small. It leads emitted photon beam to have better monochromaticity and collimation except for having larger emitted energy and higher laser utilization efficiency. We affirm the robustness of stepped focusing mirror reflecting regime through various kinds of numerical simulations.