# Fiber Mode Solvers

## Introduction

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…

## Real-valued formulation

The time independent Maxwell curl equations with a positive time convention (ej ω t ) are and the divergence equations are The factor j does not suit the current purpose, so make the substitution where n0 =  is the free space impedance, so that h and E have the same units. Then where . Eliminating the h gives the…

## Debye Potential

For finding electromagnetic fields in a cylindrical geometry it is convenient to use cylindrical coordinates r, θ and z, and Debye potentials parallel to the axis of rotation,   It is supposed that the permittivity is constant in the region in which the above equations apply. The fibers are defined as a series of concentric layers of…

## Separation of Variables

The potentials themselves are solutions of the scalar Helmholtz equation, and the particular solution is found by observing the boundary conditions imposed by physical considerations on E and h. The potentials are supposed to form modes, so a solution where the variables are separated is appropriate and a similar expression applies for the other Debye potential, ϕ…

## Solution of the Linear System

The order of the components in the vector f is chosen to help factor the matrix in (9) into two matrices as Ff = Dc where c = [ A, B, C, D ]T  , The advantage of this factoring is that it is easy to find the expressions for F–1  and D–1 . F–1  is found by…

## Dispersion Equation

The matrix to describe the whole multilayer can be written as one 4×4 matrix, the product of all layers, one layer of which is shown in (21). The dispersion equation is usually constructed by setting the determinant of this 4×4 matrix product to 0, but this is not a good choice from the numerical point of…

## LP Modes

Linearly Polarized modes can be found with the same style of approach as above, by a transfer matrix analysis of multilayer fibers.In the LP analysis, the initial excitation is assumed to have a linear polarization, and so a modal analysis supporting only one field component is used. It can be shown this is a good…

## References

           K. Okamoto, Fundamentals of Optical Waveguides, (Academic Press, San Diego, 2000).            C. Tsao, Optical Fibre Waveguide Analysis, (Oxford University Press, 1992), Part III            C. Yeh and G. Lindgren, “Computing the propagation characteristics of radially stratified fibers: an efficient method”, Applied Optics 16(2) p483-493 (1977)            J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix…