We present the observation of whispering-gallery modes (WGMs) in cylindrical micro-cavity coupled to a fiber taper based on evanescent field with a coupling efficiency of about 10%. All the WGMs spectra of cylindrical micro-cavities with four different diameters are precisely indexed by an explicit asymptotic formula with the positions and separation as parameters. By introducing the parameter “mode field radius" of WGMs, the mode separations calculated by mode field radius agree well with the experimental results. And by determining the difference between experimental resonant wavelengths and the theoretically assigned ones, the dispersive curve of quartz fiber is corrected over the spectrum range of 1295—1320nm and the precision of refractive index determination is as high as 10-5.