In reality we don't have perfect PEC but the metal we use to create Waveguide is almost very good metal; and the field is almost zero (with very good approximation) in its walls. So these modes would exist in reality by good approximation.

the Method of Variation is a good analytical example for comparison between perfect PEC and metals.

Let's assume the waveguide is perfectly conducting. Are TE and TM modes rigorous solutions or are they still approximations? If they are rigorous solutions, how do we prove this?

I do believe in uniqueness theory; so if these modes are exact solution of Maxwell's equation, they are exact modes that can be in these structures, like TE or TM in rectangular waveguide.

The field must meet the boundary condition forced by PECs; and in these structures the PEC walls are in one coordinate direction; so in texts we find that certain field components are set to zero.

I found Variational technique introduced in chapter 7 and 8 of "Time-Harmonic Electromagnetic Fields" by Roger F. Harrington proved these waveguide modes for any channel waveguides

TE and TM (and I guess TEM for multiple insulated metals) modes are a part of the solution set (think of it like basis functions). The sum of all these modes is the exact solution given boundary conditions. To a very good approximation the fields in the metal are 0, thus by the boundary conditions the surface tangential fields must also be 0 inside the insulator. The skin depth goes like ~1/sqrt(f*sigma), so for high frequency and high conductivity the thickness of penetration of "non-zero tangential fields" is very small. For copper at 1 GHz, the skin depth is ~ 2 microns. For most FEM and FDTD simulations, this thickness will be much smaller than the spatial resolution.

TE and TM modes in cylindrical waveguides are not only sotutions of Maxwell equations, they really exist. They can be used to built transmission lines, couplers, filters, T-junctions etc. If one needs a proof of these modes existance it is enough to excite e.g. rectangular waveguide with the generator of the frequency located in the range of single mode opeartion (TE10 mode) and to measure phase velocity and group velocity at this frequency. Then one can check the accuracy of theoretical solutions and be convinced that TE (at least) modes exist physically.

TE, TM or TEM modes are independent (at least in the case of a lossless waveguide). The propagation of a TM mode (in linear waveguide, eg. composed of linear media) does not depend on the fact that there is or is not the TE one at the moment. 

The idea is not to set longitunal E or H to zero. The important fact is that the modes ore orthogonal, eg. not related. One can easily describe one mode regardless of the presence of the other mode.

Yes I think the answer from Sterling Beeson is correct. It should be possible to derive a more precise model of TE or TM fields taking into account skin depth in the walls and compute the values of the fields at the boundaries. However, has anyone ever experimentally measured a skin depth...?

Dear Dr. Rumpf,

As far as I can tell, Jackson gives a rigorous derivation of the characteristics of the TE/TM modes of an infinite straight waveguide of arbitrary cross-section. Section 8.3 of his book (third edition)

http://www.amazon.com/Classical-Electrodynamics-Third-Edition-Jackson/dp/047130932X

is a good starting point.

It seems that the only assumption he makes is perfectly conducting waveguide.

In a nutshell, the vectorial field problem is reduced to a scalar problem (differential equation + boundary condtions). From the scalar solutions, the two kinds of vectorial solutions arise. Each of which has some different set of components of the electric and magnetic field that turns out to be zero.

Similar to Z Skvor's answer, it's useful to see that the TE & TM modes in a channel waveguide are independent and really don't couple (unless you specifically design your waveguide to couple them). With variation of the polarization of the input light, you can actually see the TE & TM modes exist & propagate separately (with an IR camera on the output of the waveguide). They don't really exchange power, and the images you get on the camera are exactly like those in a textbook.

So in that sense they really exist as separate modes of the waveguide and are orthogonal (exist independently), not just a mathematical trick.

This thread has aged now so I will offer my explanation that TE and TM modes do exist, but only in special cases.  First, it is possible rearrange Maxwell's equations so as to put Ex, Ey, Hx, and Hy completely in terms of just Ez and Hz.  This can be done rigorously without making any approximations.  Second, in waveguides where the fields are contained completely inside a single linear, homogeneous, and isotropic (LHI) material, the wave equation decouples so that all the field components can be solved separately, including Ez and Hz.  While Ez=0 or Hz=0 are trivial solutions, they are still perfectly valid and correct solutions.  So, we are able to solve Ez while setting Hz=0 (TM modes), and to solve Hz while setting Ez=0 (TE modes).

After giving this some thought, one can conclude that:

1. It is only possible to contain all the fields in a single LHI material when there also exists metals.  So, metals are a requirement for TE and TM.

2. The PEC approximation must be valid.  This probably breaks down at very high frequencies or very thin metals.

3. Structures like microstrips have electric fields that slice through air and the dielectric substrate so they do not support TE and TM modes.  However, since most the power is contained under the strip and inside just the dielectric, TE and TM can be a good approximation.  This is usually called quasi-TE and quasi-TM.

Along this same theme, I have read that TEM waves only exist in transmission lines with LHI materials.  I have not looked at nonlinear materials, but I have looked at transmission lines embedded in inhomogeneous and anisotropic materials and I can say that TEM waves still exist in this case.  There is just no way to produce a longitudinal component when the frequency is zero.  This may not be the case inside bi-isotropic or bi-anisotropic materials, but I have not investigated this.

Admittedly, I do not agree with the previous comment by Raymond Rumpf. TE and TM modes can be excited in a dielectric sheet, even in the case there is no metallization.

Therefore, there is no need for a PEC condition. Metals are not necessary for TE nor TM modes (but are necessary for TEM ones).