Dielectric cylinders with inhomogeneous distribution of dielectric constants are very useful for high-capacity optical communications as optical fibers with small dispersions. The measurements of refractive index distribution are indispensable for the development of manufacture of long-distance optical glass fibers with high qualities. In order to develop these scattering pattern methods, this paper discusses exact analysis of asymptotic scattering and diffraction fields of vertically incident optical waves by inhomogeneous dielectric cylinder with arbitrary graded index profile. The scattering and diffraction fields are represented by a summation of fast convergent residue value series and the steepest descent integrals, by Watson transformation, that correspond to creeping waves propagating along boundary surfaces of the cladding and direct refracted and reflected waves in core and cladding, respectively. The physical meaning of each term is studied, comparing scattering pattern experiments of graded index fibers with arbitrary inhomogeneous index distributions.