The Meaning of TM11 and TE10/TE01 Modes

[freesnow]

Why is the lowest order TM and TE mode TM₁₁ and TE₁₀(or TE₀₁)? What is the physical meaning of the different orders of the modes?
Thanks.

 

[Born2bwire]

I'm going to guess this belongs in the homework section and definitely need more information needs to be provided, like what is the waveguide?

 

[freesnow]

no, this is not homework, my lecture notes said the lowest order needs to be TM11 and TE10(or TE01), but I don't really understand it.

 

[gnurf]

If you consider the expression for the superposition of two uniform plane waves propagating symmetrically with respect to the z-axis, you will see that it contains a factor in the form of $$sin(\beta x \ cos\ \theta)$$. This factor describes the the standing wave character (here, in the x-direction). When this factor is zero (i.e., when $$\beta x \ cos\ \theta =\ m\pi$$ where $$m = 0, 1, 2, 3, ...$$) the electric field is, of course, also zero.

This is interesting because it means we can place two perfectly conducting sheets in the planes $$x = 0$$ and $$x = m\lambda /(2\ cos\ \theta)$$, without violating the boundary conditions (i.e., zero tangential electric field etc).

The fields will have m number of one-half apparent wavelengths in the x-direction between the plates.

This line of reasoning can be extended to a three-dimensional case where a quick glance at the field expressions for TE and TM waves will reveal why TM_1,1 and TE_{1,0 or 0,1} are the lowest possible modes. Plug in m=n=0 for in the expression for TE waves or m = 0 or n = 0 for TM waves and see what happens.