# 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, [2] 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

[1] K. Okamoto, Fundamentals of Optical Waveguides, (Academic Press, San Diego, 2000). [2] C. Tsao, Optical Fibre Waveguide Analysis, (Oxford University Press, 1992), Part III [3] C. Yeh and G. Lindgren, “Computing the propagation characteristics of radially stratified fibers: an efficient method”, Applied Optics 16(2) p483-493 (1977) [4] J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix…