Have you a reason for claiming that cylindrical dielectric waveguides, i.e. optical fibers, cannot support TE or TM modes?

There is an extensive discussion in chapter 12 of A.W Snyder & J.D Love "Optical Waveguide Theory", Chapman & Hall (1983). 

There is an earlier treatment in A W Snyder "Asymptotic Expressions for Eigenfunctions and Eigenvalues of a Dielectric or Optical Waveguide", IEEE Trans. Microwave Theory & Techniques, vol 17, no. 12, Dec 1969, pp 1130-1139.

For example the "quasi-LP" LP11 modes are actually a combination of hybrid and transverse HE21, TE01 and TM01 modes.

http://www.invocom.et.put.poznan.pl/~invocom/C/P1-9/swiatlowody_en/p1-1_1_3.htm

http://ieeexplore.ieee.org/abstract/document/1127112/

I suppose my thinking is that since a cylindrical dielectric waveguide is not homogeneous, there is no way to split Maxwell's equations into two distinct wave equations (usually Ez and Hz).  Following this line of thinking, Ez and Hz must exist simultaneously and thus no distinct TE and TM modes can exist, only hybrid modes.  I think in the long wavelength limit when the modal index tends toward zero, Maxwell's equations do split, but this is not a physically real mode, although maybe a good approximation.

Thank you for the references.  I will look these over.  Please correct my line of thinking!

Raymond,

For a step profile fibre, the core and cladding are individually homogeneous, with a step discontinuity at the core-cladding boundary.  There are TE solutions where the electric field is parallel to the interface at the boundary, and Ez is zero everywhere.  Similarly there are TM solutions where the magnetic field is parallel to the boundary at the interface.  In a plane wave description of such modes, the rays are meridional, passing through the waveguide axis, and preserving either Ez=0 or Hz=0 at every reflection.

TE and TM modes are also supported in a cylindrically symmetric graded profile fibre, where grad (ln n2) is orthogonal to the fibre axis.

https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-23-13-17330&id=321201

 If my revised understanding is correct, it is only circularly symmetric dielectric waveguides that would support TE and TM modes.  Since a circularly symmetric dielectric waveguide is uniform in the azimuthal direction, Maxwell's equations can split. In anything else, Maxwell's equations do not split and only hybrid modes exist.

Is this your understanding?

Raymond, "it is only circularly symmetric dielectric waveguides that would support TE and TM modes."

Discounting planar waveguides which also support TE and TM modes, I believe this is correct, but it is not something I have studied deeply.

In typical optical fibres, the LP approximation is pretty good most of the time.

Manufacturing imperfections, external bending and microbending stresses result in coupling between the almost degenerate TE, TM and HE components of an LP mode, so there is usually significant evolution of the polarisation state after propagating only a few metres in fibre.

In a way, I think I am seeing circularly symmetric waveguides similar to a slab waveguide in that they both have a uniform axis which allows Maxwell's equations to decouple into separate modes.  

I very much appreciate the input and I will be modifying the map accordingly.

Thank you!!!

I have revised the map per this conversation along with some other tweaks.

A hollow rectangular metallic wave guide in which one has laid a plane dielectric slab which is longitudinally homogeneous and which is normal to a wave guide's transversal axis does propagate modes that are TE & TM to that axis, Raymond; check Collin's book if you will.

@Denis Dang, good catch.  Indeed having one direction uniform will support TE and TM modes.  I will revise the map accordingly.

You're welcome, Raymond. It is quite a challenge, mind you, as an exhaustive graphic might call for more than 2 dimensions. Regardless, it is a good initiative which will be helpful to both the student and the engineer.

You have missed out Twin Feeder also known as Lecher lines (TEM) .  You might include sectoral (TE, TM) conical (TE, TM) and bi-conical (TEM) transmission line?  Tapered co-ax?  

In the single-conductor waveguides' category, you might want to quote the Sommerfeld line, also called the Goubau line, Raymond. It consists of a bare metal wire. It has gained popularity amongst the engineers working at THz frequencies, in the last few years, because of its low loss / dispersion in comparison to the dielectric wave guides.

A variant of that line, often called the Goubau line like the above, has a dielectric coat. On the one hand, it is more lossy and more dispersive than the bare wire. On the other hand, it allows for the travelling energy to be closer to the wire.

I really like all these suggestions.  The purpose of this diagram is just to map out waveguide categories.  However, perhaps there is a need for sort of a periodic table of waveguides.  I will start thinking about that.