I do not believe your reflection and transmission coefficients are correct.

R = (Zi  - Z0) / (Zi + Z0) is the reflection coefficient, for field amplitude, not power.  At normal incidence:

• The electric field (E) transmission coefficient is TE = 2 Zi / (Zi + Z0)
• The magnetic field (H) transmission coefficient is TH = 2 Z0 / (Zi + Z0)
• Power transmission (1-R)2 = TE TH = 4 ZiZ0 / (Zi + Z0)2

In general for your plane wave approximation, both transverse Ey and normal Hx fields are discontinuous at the {y, z} boundary between media 1 and 2. 

I am not aware of an analytic solution, but I would expect some perturbation of the otherwise spatially uniform displacement currents in the vicinity of the boundary, such that some of the incident plane wave energy is scattered away from the line of intersection between the 3 homogeneous regions.

Dear Alan, thank you for pointing out my error concerning R: I forgot to write the exponent 2. With the corrected (power) R, our field transmission coefficients and the power transmission coefficient are identical. (But I have to admit that your equations look more familiar than mine.)

Thank you, too, for referring to the discontinuous Ey! I stopped thinking at div B != 0 but yes, the introduction of a z component of curl E would be equally problematic.

It would be sad if there were not a rigorous solution at all because sometimes such solutions resp. their consequences are real eye-openers! I agree that some of the incident energy will probably be scattered by the intersection line but I don't quite see how the transmitted wave develops while propagating further into medium 1 and 2, with the cylindrical wave generated by the scattering becoming weaker.

Qualitatively, the problems with both div B and curl E would be solved if a Bz component would appear at the boundary between medium 1 and 2 ...

From your experience, do you think that using a simulation software like CST might be helpful in this case?

Since the wave is travelling parallel to the interface, I do not think any transmission or reflection equation would help.  I am also not sure how you have in mind launching the wave.  You could not numerically excite this with a pure plane wave because that is not a valid electromagnetic solution.  Plane waves only exist inside homogeneous media.  

One could envision a situation where a plane wave in air is incident onto a surface where one half is medium 1 and the other is the second half is medium 2, each having a different impedance.  Certainly the beam would become asymmetric and most likely not propagate well along the interface at long distances.  The wave would partly scatter after incidence and would probably also tend to diverge away from the interface as the beam continues to propagate.  Reflection from the interface would also be asymmetric, likely reflecting stronger from the lower impedance medium.  Given both media have the same refractive index, I would expect this divergence to be slow since each half of the beam would remain in phase.  I think the electric field would be more intense in the high impedance medium and the magnetic field would be more intense in the other region.

This configuration would probably only take a few minutes to simulate.

 Dear Raymond, the situation you describe in the first sentence of your second paragraph is exactly what I meant with the text and picture in the file attached to my question ("TwoMedia.pdf"). Of course, plane homogeneous waves are just approximations for the sake of simplicity. One could replace the plane wave by a spherical wave; in this case, each medium would have the form of one half of a spherical shell of a very large radius and large thickness.

Thank you for your thoughts on the way the two parts of the wave will propagate along the boundary! I tend to agree but I'm still not capable of visualizing the details; after all, the problem posed by "div B != 0" and "z component of rot E" (if the wave were just split into halves) has to be solved immediately once the wave has entered the media. That's the reason I would dearly love to see an analytic solution.

Last year I faced a similar problem with cylindrical waves (another approximation): Only after finding a rigorous solution in a book by Weeks, I realized that waves can be reflected by empty space! I would never have dared to suppose something like this just by qualitative reasoning.

If no hint to an analytic solution appears within the next weeks, I will follow your advice and try a simulation.

What do mean by waves reflecting off of empty space?

If I get time, I will run a quick simulation.  That will not be the analytical solution you are seeking, but may guide you.

The attached picture is a close-up of the fields Ez and Hphi of a cylindrical wave according to equations (10) and (11) in my working paper "On Divergence in Radiation Fields". The (unrealistic) line radiator is of unlimited length, surrounded by vacuum resp. air, and carries a current of equal phase everywhere (phase independent of z). The distance from the radiator is 10 * wavelength. Since these are cylindrical coordinates, different signs of the fields mean outward power flow (as in the unipolar pulses) while equal signs are related to an inward flow (between the pulses).

Perhaps this is less surprising if we reduce the (unlimited) length of the line radiator, and terminate its ends by perpendicular PEC surfaces as mirrors. If we look at these parallel mirrors as a wave guide, we see that its impedance for a cylindrical wave reduces with increasing r. And another point is that without the backward flow of energy, the fields would not comply with the Maxwell equations.

Many thanks in advance for your willingness to run a simulation! Provided you get time, I'm curious about the outcome!

Well, I went ahead and simulated this using the finite-difference frequency-domain method.  This is a rigorous method so no approximations were made to Maxwell's equations.  Attached are the results for two different polarizations.  Wave is incident from the top.

Raymond, this is great, and you were much faster than I dared to hope! Thank you very much! The first thing that catches my attention is the scattering by the line where all three media meet, as you and Alan predicted.

As to be expected, the E field is stronger in the high impedance medium, and the H field at low Z.

But I'm surprised by the pictures of the y component! In the TM case, I expected that along the boundary of medium 1 and 2 a Hy component would be generated during entering from air, and would remain stable with increasing y in order to "terminate" the stronger Hx field in the low Z medium. However, Hy (TM) seems to become weaker with increasing distance from the scattering line while the difference between Hx in the low Z and in the high Z medium seems to be independent of y. To put it another way, the diverging away from the boundary seems to be completely absent (as far as one can see). Would you interpret the results differently?

I guess the white lines at the boundaries are just graphical items without physical reality. Or are the media indeed separated by a short distance?

The white lines are the interfaces between the materials just drawn on top of the fields.  There are no small separations.

Yes, I am not seeing the beam diverge from the boundary nearly as much (if at all) as I expected.

Results not quite meeting our (a bit vague) expectations is a stimulating experience in my opinion. Raymond, did you use a commercially available simulator, and if so, could you please tell me the name? I thought about using CST, but at the outset, this might be kind of overkill.

Pretty much any commercial software can do this kind of simulation.  I used my own code that implements the finite-difference frequency-domain method.  It only took me a few minutes to do (lots of practice).  If you are curious about the method, see the attached paper.  You can also take a look at Lectures 11 and 13-15 here:

http://emlab.utep.edu/ee5390cem.htm

Raymond, thank you very much for the article and the link to your lectures! With some of the CEM methods, I'm not familiar but your course seems to me to provide a pretty overview on this field. As time allows, I will enjoy it bit by bit, beyond the lectures you recommended.

Initially, I was resolved to use existing software but now I decided to write a FDFD simulation myself in order to benefit from the related freedom. In the last decade, I used mainly Mathematica; yesterday, I tried Octave but, as usual, stumbled at the outset over minor technicalities.

In the meantime, I'm still wondering about your simulation results. In a naive way, one could argue that due to total reflection, each wave "sees" a perfect continuation of itself beyond the boundary, and therefore is able to propagate unchanged (apart from the superposition by the cylindrical wave caused by scattering). But there should be an "objective" explanation based on EM waves ...