A model to calculate the butt-joint coupling efficiency between a multi-mode fiber to a single-mode fiber is presented based on a mode field coupling theory. The dependence of the coupling efficiency between the multi- and single-mode fibers on the core radius of the multi-mode fiber is investigated. A experiment is conducted with 532nm laser to measure the butt joint coupling efficiency of a multi-mode fiber to a single-mode fiber. Good agreement is obtained between the experimental results and the theoretical predictions. A cascaded multi- and single-mode fiber relay system is proposed to transport a laser beam with high power and good beam quality.