Relationship between group velocity and phase velocity of the mode-field on multi-mode fiber

To clarify the discussion on ray optics and wave optics on the delay of mode field propagation process in the multimode fiber theoretically, the relationship between phase velocity Vp and group velocity Vg is derived, Vp*Vg=(c/n1)2, according to the relationship between the mode characteristic roots U and the optic fiber normalized frequency R, as well as the weak waveguide approximation theory. Based on the results, it is concluded that phase velocity is slow and group velocity is fast at low-order mode, phase velocity is fast and group velocity is slow at high speed mode. Ray optics and wave optics are used for bandwidth estimation to a given optical fiber, and the consistency of the results of two theoretical calculations confirms the correctness of these relations.