Difference in proof between TE and TM modes

AI Thread Summary
The discussion centers on the mathematical differences between TE (Transverse Electric) and TM (Transverse Magnetic) modes in waveguide theory. For TE modes, the solutions involve specific boundary conditions leading to symmetric and asymmetric equations, while TM modes introduce the dielectric constant into the eigenvalue equations. The key distinction is that in TM modes, the magnetic field component Hz is zero, while in TE modes, the electric field component Ez is zero. The user seeks clarification on how to derive the eigen equations for TM modes, which are similar to those for TE modes but account for the dielectric constant. Understanding these differences is crucial for grasping wave propagation in waveguides.
GengisKhan
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I'm a little confused regarding the maths of TE and TM modes.

Solving the following system for TE (which derives from Ey(x, z, t) = Em(x) = exp[i(ωt-βz)] ):
Asin(px) + Bcos(px) , -d/2 < x <d/2
Cexp(-qx), x>d/2
Dexp(qx), x<-d/2

we conclude in two types of solutions for TE modes: symmetric: ptan(pd/2) = q and asymmetric: pcot(pd/2) = -q.

What is different for the above solutions for TM? I have a hard time determining that. I think it has something to do with the dielectric constant, but I'm not quite sure.

I can elaborate on any maths you ask for. Also sorry for the quality of my post, it is my first one.
 
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Hi, when you write the solution Ey =exp[i(\omega t - \beta z)], this is a wave traveling in the negative z direction.

If you want to break this wave as the sum of a TE mode and a TM mode, then for the TE wave, only Ez is 0. Ex, Ey, Hx, Hy, and Hz are not zero. for the TM wave, Hz is 0. Ex, Ey, Ez, Hx, Hy are not zero.

I cannot relate your question to the definitions of the TE and TM mode.

elgen
 
Elgen, thanks for your answer! That's the very first part of the proof, I'm trying to understand beyond that.
Fortunately, I found some insight from http://ece562web.groups.et.byu.net/notes/slab_waveguide.pdf.
TM Modes solutions are found at pages 8-9. The problem is that I don't know how to reach the eigen equations, which are the same as TE's (described earlier in the .pdf), only with the dielectric constant n^2 added. If someone can briefly describe what is different from TE, I'd be grateful!
 

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