Scattering characteristic of an inhomogeneous cylindrical particle plays an important role in the measurement of parameters of complicated particles. Based on Debye theory, scattering by an infinite multi-layered cylinder in an on-axis Gaussian beam is discussed in detail. Scattering coefficient formulas of Debye series are obtained. With the formulas, distributions of total scattering intensities, Debye component scattering intensities by a homogeneous and a two-layered cylinder are calculated. And the results of total scattering intensities are in line with those of generalized Lorentz Mie theory. It is indicated that the different scattering angle interval of total scattering intensities comes from the contribution of different-order scattering intensities. Presence of two maximums in two order scattering intensities of Debye series is determined by radius and refractive index of a two-layered cylinder. For a two layered cylinder with a thin shell, there are two maximums between 120° and 150°, which is called dual first-order rainbow. Otherwise, when the external layer is thicker than the inner one, there is only one maximum.