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Optical Waveguide Field Equations

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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



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:








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







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.

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