The relation can be equivalently described by b-V relation. The b-V diagram of a step index optical fiber for different modes is shown in the following.

As can be seen clearly, the b-V diagram is not linear and consequently, there is dispersion.

The waveguide dispersion due to non-linear b-V diagram is obtained as

From the b-V diagram we can calculate , and subsequently the waveguide dispersion.

The peaks around V=1.2 for the lowest order mode. Consequently, in the single mode optical fiber the dispersion is maximum around V = 1.2.

For higher V-number the dispersion is low. However, as pointed out earlier, for the fiber to be single mode, V < 2.4.

The single mode optical fibers therefore have V-number very close to but less than 2.4 so that they have lowest possible waveguide dispersion.

The figure shows the plot of material, waveguide and total dispersion.

Note: The material dispersion is zero at 1270nm. However, in a single mode fiber, we always get dispersion which is a combination of material and waveguide dispersion. Therefore

The dispersion is positive for wavelengths above 1310nm and negative below 1310nm.

Note : The negative dispersion does not mean pulse compression. Due to dispersion the pulse always broadens and the broadening is

Where L is the length of the fiber and is the spectral width of the source.