Based on the transformation theory of ray matrix and the overlap area integration method, a theoretical analysis has been performed and the coupling efficiencies of optical signal have been calculated in single-mode fiber, expanded fiber, laser diode, and optical waveguide etc, in taking into consideration of specific boundary condition in each case. The obtained formulation is applicable in many practical issues. A manufacture method of expanded fiber is presented and described with our newly formed theory. A comparison of theoretical with experimental results was made.