EM waves: multiple planar interfaces (PEC-backed lossy dielectric)

[bladesong]

Homework Statement

Our prof has told us we can get help from wherever/whoever we want as long as it isn't classmates. This is a take-home test. The relevant question:

You have a three-layer dielectric.

| Layer 1 | Layer 2 | Layer 3 |

Layers 1 and 3 are semi-infinite.

Layer 1 is air (ε_r = 1, μ_r = 1), σ = 0.
Layer 2 is a lossy dielectric, ε_r = 2, μ_r = 1, σ = 0.01S/m. Its length is 1m.
Layer 3 is a perfect electric conductor (PEC).

The freespace wavelength of an incident EM wave is 1m.

Find the power reflection coefficient R's numerical value for an incident angle of 0°, 40°, and 80°, and plot R for 0<θ<90.

Homework Equations

Maxwell's equations, boundary conditions, and matrix methods for solving multiple dielectric interfaces. I've posted some, the rest I can link to or post them on request: they're quite lengthy (probably about 20-30 pages).

The Attempt at a Solution

I've tried to find a closed form for the transfer matrix between the lossy dielectric and the PEC, but it always comes up singular. I'm not sure how to procede. As the σ = ∞ for the PEC, the wavenumber is also infinite, and so I can't reasonably find a closed form for the matrices. I have done a numerical solution that literally measures reflections (and attentuation causes it to quickly drop off anyway), but I'm wondering if I was beating my head against the desk for nothing.

If someone could explain what a PEC does in this case (does it absorb all the energy? Does it completely reflect it?) it would be really helpful, information on that seems scant too.

I can, again, post my attempt if you wish, but it's about 300-400 lines of MATLAB code. If in the end the answer is just "there is no closed form for the matrix" then I guess that'll be that.

Thanks and if there are any additional questions please don't hesitate. You guys are the best, that's why I keep coming back.

 

[mark726]

Hello,

Thank you for sharing your question and attempt at a solution. It appears that you are on the right track in using Maxwell's equations and boundary conditions to solve for the power reflection coefficient R in this scenario.

Regarding your question about the PEC, it is important to note that a perfect electric conductor is an idealized material that has an infinite conductivity and therefore, does not allow any electric field to penetrate into it. This means that for an incident EM wave, all of the energy will be reflected back. In other words, the PEC acts as a perfect reflector.

As for your difficulty in finding a closed form for the transfer matrix between the lossy dielectric and the PEC, it is possible that there may not be a simple solution due to the complexity of the problem. In this case, your numerical solution may be the best approach to obtain the power reflection coefficient for different incident angles.

Overall, it seems that you are on the right track and have a good understanding of the concepts involved. Keep up the good work and don't hesitate to ask for further clarification or assistance if needed. Good luck with your solution!