# Optical Waveguide Field Equations

This is a continuation from the previous tutorial - Waveguide Modes.

For a linear, isotropic dielectric waveguide characterized by a spatial permittivity distribution of ε(x, y), Maxwell's equations in (40) and (41) [refer to the Harmonic Fields tutorial] can be written as

$\tag{8}\pmb{\nabla}\times\mathbf{E}=-\mu_0\frac{\partial{\mathbf{H}}}{\partial{t}}$

$\tag{9}\pmb{\nabla}\times\mathbf{H}=\epsilon\frac{\partial{\mathbf{E}}}{\partial{t}}$

Because the optical fields in the waveguide have the form of (1) and (2) [refer to the waveguide modes tutorial], these two Maxwell's equations can be written in the following form:

$\tag{10}\frac{\partial{\mathcal{E}_z}}{\partial{y}}-\text{i}\beta\mathcal{E}_y=\text{i}\omega\mu_0\mathcal{H}_x$

$\tag{11}\text{i}\beta\mathcal{E}_x-\frac{\partial{\mathcal{E}_z}}{\partial{x}}=\text{i}\omega\mu_0\mathcal{H}_y$

$\tag{12}\frac{\partial{\mathcal{E}_y}}{\partial{x}}-\frac{\partial{\mathcal{E}_x}}{\partial{y}}=\text{i}\omega\mu_0\mathcal{H}_z$

and

$\tag{13}\frac{\partial{\mathcal{H}_z}}{\partial{y}}-\text{i}\beta\mathcal{H}_y=-\text{i}\omega\epsilon\mathcal{E}_x$

$\tag{14}\text{i}\beta\mathcal{H}_x-\frac{\partial{\mathcal{H}_z}}{\partial{x}}=-\text{i}\omega\epsilon\mathcal{E}_y$

$\tag{15}\frac{\partial{\mathcal{H}_y}}{\partial{x}}-\frac{\partial{\mathcal{H}_x}}{\partial{y}}=-\text{i}\omega\epsilon\mathcal{E}_z$

From these equations, the transverse components of the electric and magnetic fields can be expressed in terms of the longitudinal components:

$\tag{16}(k^2-\beta^2)\mathcal{E}_x=\text{i}\beta\frac{\partial{\mathcal{E}_z}}{\partial{x}}+\text{i}\omega\mu_0\frac{\partial{\mathcal{H}_z}}{\partial{y}}$

$\tag{17}(k^2-\beta^2)\mathcal{E}_y=\text{i}\beta\frac{\partial{\mathcal{E}_z}}{\partial{y}}-\text{i}\omega\mu_0\frac{\partial{\mathcal{H}_z}}{\partial{x}}$

$\tag{18}(k^2-\beta^2)\mathcal{H}_x=\text{i}\beta\frac{\partial{\mathcal{H}_z}}{\partial{x}}-\text{i}\omega\epsilon\frac{\partial{\mathcal{E}_z}}{\partial{y}}$

$\tag{19}(k^2-\beta^2)\mathcal{H}_y=\text{i}\beta\frac{\partial{\mathcal{H}_z}}{\partial{y}}+\text{i}\omega\epsilon\frac{\partial{\mathcal{E}_z}}{\partial{x}}$

where

$\tag{20}k^2=\omega^2\mu_0\epsilon(x,y)$

is a function of x and y to account for the transverse spatial inhomogeneity of the waveguide structure.

The relations in (16) - (19) are generally true for a longitudinally homogeneous waveguide of any transverse geometry and any transverse index profile where ε(xy) is not a function of z. They are equally true for step-index and graded-index waveguides.

In waveguides that have circular cross sections, such as optical fibers, the x and y coordinates of the rectangular system can be transformed to the r and φ coordinates of the cylindrical system for similar relations.

Therefore, in a waveguide, once the longitudinal field components, $$\mathcal{E}_z$$ and $$\mathcal{H}_z$$, are known, all field components can be obtained. The fields in a waveguide can have various vectorial characteristics. They can be classified based on the characteristics of the longitudinal field components:

1. A transverse electric and magnetic mode, or TEM mode, has $$\mathcal{E}_z=0$$ and $$\mathcal{H}_z=0$$Dielectric waveguides do not support TEM modes, as can be seen from (16) - (19).

2. A transverse electric mode, or TE mode, has $$\mathcal{E}_z=0$$ and $$\mathcal{H}_z\ne 0$$.

3. A transverse magnetic mode, or TM mode, has $$\mathcal{H}_z=0$$ and $$\mathcal{E}_z\ne 0$$.

4. A hybrid mode has both $$\mathcal{E}_z\ne 0$$ and $$\mathcal{H}_z\ne 0$$. Hybrid modes do not appear in planar waveguides but exist in nonplanar waveguides of two-dimensional transverse optical confinement. The HE and EH modes of optical fibers are hybrid modes.

The next part continues with the Wave Equations for Optical Waveguides tutorial.