On the basis of the generalized Lorenz-Mie theory, a new method to study Gaussian beam scattering by a multi-layered sphere is presented by the formula of the Debye series decomposition for scattering coefficients. The results calculated by this method and by the generalized Lorenz-Mie theory agree well. The scattered field distribution in the far-zone of a multi-layered sphere illuminated by Gaussian beam is simulated and the interference intensity distribution of mixed order rainbows is separated efficiently. The refractive index profile of the multi-layered sphere is exponential versus the radial position. The intensity distribution of the twin first-order rainbow and the Airy structure of each layer by a two-layered sphere are simulated. At last, the influence of incident position and beam waist on scattering intensity distribution is discussed.