|| Since 1970, there have been many analytic theories studying waveguide modes in optical fibers. As the years progressed, the structure of optical fiber and its characteristics have undergone many changes. In recent years, the methods of analysis have also evolved into a more numerical style, such as the finite element and finite difference approach. In this thesis, we propose a semi-analytic, coupled Ez and Hz method for solving a multiple layered piecewise constant cylindrical dielectric waveguides.|
In our mathematical model, we are able to handle any arbitrary layered structure, in particular, the step-index fiber, W-type fiber, and dielectric tubes. Within each layer, we express the azimuthal field components in terms of Bessel functions whose coefficients are determined by two pairs of Ez and Hz components that define the layer. By equating the transverse fields (above) on either side of each layer interface the coupled field equations are derived. The field components are either real, or purely imaginary, this allows us to formulate our matrix in real arithmetic. Further simplification is possible by using the Wronskians of the Bessel functions. The resulting matrices are both real and symmetric, which is consistent with the reciprocity principle. Compared to the traditional formulation from the early 1970s, we have reduced the variables by half and extended the formulation to include any arbitrary number of layers.
Numerous numerical results are presented in this thesis for all three types of fiber previously mentioned. Both lower-order modes as well as higher-order modes including TE, TM, HE, and EH modes are presented and discussed. Our formulations are compared to that of textbook formulas for the simple two-layered step index fiber, and are found to be identical.