What Determines the Mode of Propagation in Wave Guides?

[quasar987]

In finding the wave solution inside a wave guide, we got that the wave number of the wave inside the guide is given by

$$k_g^2=\epsilon \mu\omega^2 -\pi^2\left[ \left(\frac{m}{a}\right)^2+\left(\frac{n}{b}\right)^2 \right]$$

where n,m can take any integer value. The pair (n,m) caracterizes the mode of propagation (i.e. the wavelenght of the propagating wave). For frequencies lower than the cutoff frequency $\omega_c_{mn}$, the wave number is purely imaginary and there is no wave propagation along the guide; just some atenuated electroagnetic "disturbance".

This is all pretty clear from a mathematical standpoint, but what physically determines the mode? Say I'm shooting a wave of a given frequency into a guide. How do I know what n and m are??

 

[marcusl]

The equation you are looking at is for a rectangular waveguide. The indices m and n of the eigenfunctions or specific modes give the number of half wavelengths present between the guide walls in the horizontal and vertical directions. In addition there are two separate types of waves, transverse electric and transverse magnetic, so there are two sets of solutions TE_m,n and TM_m,n. Every one is associated with a different frequency if a is different than b (rectangular).

Which wave you propagate depends on the frequency and the boundary conditions of the way you launch it. For instance, you can launch a TE wave with a short dipole-type stub and a TM wave with a loop. It's actually more complicated, of course, because most modes have E and B fields so you can launch a TE wave with a loop if you put it in the right place, etc. If your frequency lies between eigenmodes, I think you get a mixture of stuff.

If you want to see more, check out one of the many good books from your university EE library. Old ones tend to be best because, after WWII and the invention of radar, they were catering to a huge newbie audience wanting in-depth explanations of what was then brand new and unfamiliar microwave technology. I forget book titles, so I'll list the authors:

George Southworth invented waveguides and microwave instrumentation at Bell Labs. His book and articles are notable for plots of the fields for different modes, and for discussions and pictures of how to launch specific modes with stubs, loops, apertures, etc.

Sarbach and Edson give clear treatments of the math and reproduce a lot of Southworth's diagrams and data.

The Rad Lab books had 2 volumes on waveguides:
Marcuvitz, Waveguide Handbook (better of the two) and
Ragan, something like Microwave Transmission

Even modern books on microwave engineering will have sections on waveguides that are more extensive that physics E&M books. Robert Collin's Foundations of Microwave Engineering is one of the classics.

Edit: I might have implied that Southworth was sole inventor of waveguides. Chu at MIT came up with it around the same time (they published back to back articles) and Rayleigh had worked out the math long before.

 

[Dr Transport]

Schwinger wrote some really good reports based on his time at the Rad Lab during WWII. Slater wrote two mongraphs on microwaves and microwave electronics, one of which I have had more than one opprtunity to consult during my career. Collin's book on guided waves is also very good and recently updated.

 

[marcusl]

I was trying to think of things that would have illustrations and intuitive explanations, and are readily available in libraries. Field Theory of Guided Waves emphasizes math, and Quasar indicated he's fine with the math but is looking for physics intuition. I imagine that Schwinger and Slater fall in the same category?

 

[Claude Bile]

The equation you quoted gives the value of the propagation constant, given the mode. It tells you nothing about which modes will actually be excited when you launch a wave into the guide.

The modes that are excited will depend on the value of the overlap integral between the mode and the initial E field, i.e. how closely the mode and the initial E field 'match'.

As far as textbooks go, Marcuse's "Theory of Dielectric Optical Waveguides" is considered the bible of waveguide optics if you are interested in that 'regime' of waveguide physics.

Claude.

 

[Amr Morsi]

Hi man,

First, I want to express my admire for your question. As it reveals a very high understanding of Wave Guide Electromagnetics as well as of physics and mathematics as a whole.

Actually, many professors don't know the answer to this question.


*According to my understanding:

Modes are space spectral components of the main field such as FT (in time) gives the time spectral components (frequency components).

To sum up: all modes, of all kinds (Transverse and Single Transverse or even General) are liable to pass. And, they are actually present, even if some are under the cutoff frequency, but attenuating. The amplitude of each mode can be determined according to the EM Source spatial distribution. It can be done mathematically of course.

Welcome for any more questions.

Amr Morsi.