Log

A collection of cool stuff picked up along the way...

Solutions to Maxwell's equations give the eigenmodes of a waveguide, most commonly the TE and TM modes of the prototypical dielectric planar waveguide with metal cladding (where the longitudinal component of the E and H fields are zero everywhere).

For general optical waveguides, TE and TM modes do not necessarily exist as $$e_z$$ and $$h_z$$ are generally coupled together. The modes thus has both components and are called hybrid HE and EH modes. The physical intuition alludes to the mixing of TE and TM modes at reflections on the core-cladding interface, given an arbitrary waveguide cross-sectional profile, as explained in [Snyder84]¹⁾.

We now consider the case of circularly symmetric optical fibers. When the refractive indices of cladding and core do not differ significantly (so-called weakly-guiding approximation from [Snyder69]²⁾), the index difference between modes cannot be distinguished and results in mode coupling (between the TE, TM, HE and EH modes) to form the linearly-polarized (LP) modes, with each LP mode sharing the same effective index. [Wang21]³⁾ illustrates this rather nicely in the following diagrams:

The LP modes can generally be derived as solutions to Maxwell's (Helmholtz's) equations in cylindrical coordinates, using the weakly-guiding approximation.