Laser-Compton scattering X-ray (LCS-X) sources has been expected as a compact and powerful source, beyond X-ray tubes. It will enable laboratories and companies, opening new X-ray science. It is well known that luminosity depends on the collision angle of laser and electron beam. Head-on collision is ideal in the point of maximizing the luminosity, though difficult to create such system especially with optical enhancement cavity for laser. In collider experiments, however, crab crossing is a promising way to increase the luminosity. We are planning to apply crab crossing to LCS, to achieve a higher luminosity leading to a more intense X-ray source. Electron beam will be tilted to half of the collision angle using an rf-deflector. Although crab crossing in laser-Compton scattering has been already proposed , it has not been demonstrated yet anywhere. The goal of this study is to experimentally prove the luminosity increase by adopting crab crossing. In this conference, we will report about our compact accelerator system at Waseda University, laser system favorable for crab crossing LCS, and expected results of crab crossing LCS.