The material dispersion of a tapered fiber is described by Sellmeier's equation. The dependence of refractive index on wavelength and doping concentration is discussed. A He-Ne laser with the output wavelength of 632.8 nm is used in the experiment. When the cutoff frequency of the fiber is less than the laser frequency, the guiding modes of a single-mode fiber (at 1550 nm) are investigated. The results show that the original single-mode fiber becomes a multi-mode waveguide. The propagation and mode coupling of the light in the taper region are analyzed. By controlling the taper end size of the fiber, the unique tapered fiber can convert a multi-mode beam into a single-mode one.