Many kinds of optical fiber can be described, or in the case of graded index fibers, approximated by, a series of concentric layers of loss-less dielectric. When the index contrast in the structure is small, it is common to use the scalar wave equation to obtain the linearly-polarized modes (LP modes). However, when the index contrast becomes larger, the LP approximation becomes inaccurate, and a full vector analysis is necessary [1]. Within any one of the concentric layers, the field components are all linear combinations of Ordinary and Modified Bessel functions, and the solution is a matter of matching the tangential field components of adjacent layers, leading to the solution of a linear system. There are some choices to make about the best way to arrange the linear system (see Ref. [2] for a complete treatment) but probably the best choice for a general, quick algorithm is the method of Yeh and Lindgren [3]. The method has intuitive appeal in its similarity to the transfer matrix method used for the analysis of planar waveguides [4]. Both methods use a matrix to express the fields at one side of a layer given their values at the other side, and the whole structure is then characterized by cascading the layers by matrix multiplication. For vector fields in fibers, there is no natural separation into TE and TM polarization, so one must include the two tangential components for both electric and magnetic field simultaneously. In total there are four transverse components at the layer boundaries, so the transfer matrix in fiber is a 4×4 matrix, instead of 2×2.
The method used by the Optiwave fiber mode solvers is different from the previous works in three ways. The 4×4 matrix method for vector modes in fibers is implemented with these improvements to make the algorithm faster in execution time, and more reliable in the finding of modes, particularly in cases where the modes are almost degenerate. The first improvement involves a reformulation of the basic equations to make a real-valued numerical implementation. Since it is loss-less modes on a fiber of loss-less material that is being sought, there is no reason to include complex numbers in the formulation. Real-valued implementation uses less computer resources. Second, the complete transfer across one layer boundary requires a matrix inverse. It is better to find this inverse analytically, rather than rely on numerical inversion at each stage. The layer matrix can be decomposed into two 4×4 matrices, and the new matrices are of a form where it is easy to find the inverse by inspection, thereby finding the analytic solution to the original inversion problem. Third, it is usual to construct the dispersion function (the zeros of which are located at the modal indices) as the determinant of the 4×4 matrix system. While this is theoretically correct, it is not the most intuitive prescription. Worse, the determinant is not the most convenient construction for locating the zeros, particularly in the case where two zeros are very close to each other. Optiwave uses an eigen-value analysis that splits the dispersion function into two functions. This splitting resolves the almost degenerate pairs and helps the simple-minded computer algorithms to find the zeros of the dispersion function (i.e. the modes) more reliably.