Undergrad Waveguide Question: TE & TM Waves Explained

[pierce15]

I am a little confused about the difference between TE and TM waves in a waveguide. Let's say a monochromatic plane wave is incident on a wave guide. Then will this result in both TE and TM waves such that the sum of the guided waves at the entrance of the waveguide agrees with the plane wave?

 

[jasonRF]

I am a little confused about the difference between TE and TM waves in a waveguide.

This issue is that a waveguide does not support transverse electromagnetic (TEM) waves, where both E and B are perpendicular to the direction of propagation. So the waves that propagate must have a component of the electromagnetic field parallel to the direction of propagation. Waves for which the electric field is perpendicular to the direction of propagation are called transverse electric (TE); these waves have a component of the magnetic field along the direction of propagation. Likewise waves for which the magnetic field is perpendicular to the propagation direction are TM, and these waves have a component of E along the propagation direction. But it is also possible for a wave in a waveguide to have components of both E and B along the propagation direction, for example:

Then will this result in both TE and TM waves such that the sum of the guided waves at the entrance of the waveguide agrees with the plane wave?

Yep. In general it will excite many TE and TM modes. The wave will have E and B in all directions, including along the direction of propagation.

EDIT: note that in general there will be a wave reflected from the waveguide as well, which complicates things a little. But conceptually you have the right idea. Actually performing these kinds of calculations can be fairly messy and not so insightful, in my experience (from many years ago...).

jason

 

[pierce15]

Thanks, that helps.

An unrelated question, but one that started when I was thinking about this: is it possible to characterize the light coming out of a laser beam as a superposition of plane waves (i.e. Fourier transform of some function)? Or to handle it in some other way while still thinking about E and B?

My intuition is that if you shine a laser into a hollow cavity it seems like it would pass right through without exciting any TE or TM modes, if the width of the laser beam is less than the waveguide radius (assuming circular cylinder).

 

[jasonRF]

Yes, Fourier analysis can be useful for these types of systems.

Reagarding the laser beam, for all practical purposes the waveguide has essentially no effect. If you have ever looked through a metal pipe you know that optical frequencies have no trouble propagating. If you want the exact answer, then the waveguide will have a small effect - the extent of which depends on the details of the beam shape ans the size of the waveguide. I suspect it will be unmeasurable in many cases, though.

 

[Andy Resnick]

Thanks, that helps.

An unrelated question, but one that started when I was thinking about this: is it possible to characterize the light coming out of a laser beam as a superposition of plane waves (i.e. Fourier transform of some function)? Or to handle it in some other way while still thinking about E and B?

Within a resonant cavity, the EM field consists of stable modes: both longitudinal and transverse. The laser output consists of coupling the internal modes to the outside via a partially reflecting mirror. The usual way to describe the emitted field is to decompose the emitted beam in terms of the cavity modes- most laser output is a single transverse mode (e.g. a Gaussian beam), but higher-order modes (Laguerre-Gaussian or Hermite-Gaussian, depending on the cavity cross-section) are possible as well. Other, more complicated cavities (unstable resonators, for example) have different eigenmodes.

https://courses.engr.illinois.edu/ece455/Files/Galvinlectures/02_CavityModes.pdf

One helpful fact is that the Fourier transform of a Gaussian function is another Gaussian- making it trivial to perform some basic optical analysis.

 

[pierce15]

Thanks to both of you.