Calculation of parameters (propagation constant, phase and group velocities) of a graded-index optical fiber

For a weakly guiding, single-mode, graded-index circular optical fiber, the general form of the dependence of the propagation constant on the waveguide parameter is obtained. From Maxwell's equations, an equation for the field in a light guide with a gradient refractive index profile is derived. Using a power-law refractive index profile for the first three powers and a Gaussian index profile as examples, dependences of the propagation constant, phase and group velocities on the waveguide parameter are obtained. For the ratio of the power transferred by the mode to the total stored energy per unit length of the waveguide, a dependence on the waveguide parameter is plotted. It is shown that as the waveguide parameter increases and the degree of the power-law profile increases, the fraction of transferred power decreases and approaches the fraction of transmitted power for the Gaussian profile. The results obtained can be used to create waveguides for specific applications.