The complex source point method and vector mode function expansion are used to calculate the electromagnetic beam scattering by the Luneberg lens. A six-layered medium with different radii is considered as a model of the Luneberg lens. A comparison between the well-known numerical result in the Cartesian coordinate system and the new numerical one in the vector mode function is made for the incident Gaussian beam propagation in free space. The propagation properties are in good agreement with each other. The total electric field distributions along the transversal and longitudinal axes are examined for scattering by the Luneberg lens, whose dimension is assumed to be comparable with the beam width. The focusing effect of the Gaussian beam is much lower than that of the plane wave. The spot size focused by the Luneberg lens almost equals the wavelength for the various beam parameters including the plane wave.