Why can we see light when looking down a metallic tube? TEM modes can't exist.

[Kyle91]

Hey everyone, I'm going over my course content for an exam this Saturday and I'm a bit confused by a realisation I've just had.

In a metallic waveguide we know there are no TEM modes. But wouldn't this mean that we couldn't see through metallic waveguides as we see electromagnetic radiation? Obviously this only applies when the waveguides are perfect conductors, but I don't think this is the reason either because we can assume it's at least a 50% perfect conductor and thus should lock 50% of the light.

Any insight?

 

[Simon Bridge]

You mean like these things?
"a radio wave in a hollow metal waveguide must have zero tangential electric field amplitude at the walls of the waveguide, so the transverse pattern of the electric field of waves is restricted to those that fit between the walls. For this reason, the modes supported by a waveguide are quantized. The allowed modes can be found by solving Maxwell's equations for the boundary conditions of a given waveguide."
... you can think of it as like how standing waves on a string fixed at each end must have zero tengential amplitude at the end - the electric field in a metal tube must show an antinode at the walls. That's all the "no TEM mode" bit means iirc.

 

[Kyle91]

I don't think I communicated clearly. Look on that same wikipedia page under Analysis, it says -

"TEM modes (Transverse ElectroMagnetic) have no electric nor magnetic field in the direction of propagation."

This is due to -

If we sub in E_z and H_z = 0 (being transverse Electromagnetic Waves) then E_x = H_x = E_y = H_y = 0. And so there cannot exist a transverse electromagnetic wave in a perfectly conducting waveguide.

The only reason I can think of as to why we can see light going down metallic tubes is that it comes in at a slight angle, but I'm not by any means confident with this explanation.

 

[Simon Bridge]

If we sub in Ez and Hz = 0 (being transverse Electromagnetic Waves) then Ex = Hx = Ey = Hy = 0. And so there cannot exist a transverse electromagnetic wave in a perfectly conducting waveguide.

And yet - clearly, there can - since EM waves go through waveguides all the time. Further, you know that this result is completely compatible with maxwells equations. It follows that you are misunderstanding them, or you are misunderstanding what you are being told when it is asserted that TEM modes are zero.

The requirement that the transverse amplitude is zero at the surface of the guide means that some wavelengths won't make it out the other end of the tube. Somewhere in your notes there should be a derivation of the EM field in a waveguide with conducting walls.

 

[Kyle91]

So what are you saying is wrong? Those formulas clearly show that EM waves propagating in z cannot exist - assuming perfectly conducting, propagation solely in z etc. etc.

 

[vanhees71]

It's correct that there are no TEM modes in waveguides with simply connected cross sections, but how do you come to the conclusion that no waves can go through it? There are many propagating TE and TM modes!

 

[Simon Bridge]

Those formulas clearly show that EM waves propagating in z cannot exist

This is what I am saying is wrong - that is not what the formulas are saying. You are making a mistake. I'm trying to get you to elaborate on your reasoning so the nature of the misunderstanding will be exposed.

Of course you could just solve maxwels equations for the EM wave in a metal guide and see that you do, indeed, get waves out the end. But I suspect that won't clear up your confusion.

aside: Thanks vanhees71 :)

 

[Kyle91]

Yeah I know that Transverse Electric (TE) and TM modes can exist in these type of waveguides but I was under the impression light we see has to have an Electric and Magnetic component.

Does that mean that we can see EM waves with no M or no E field?

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Edit: I should have been more clear, I meant "Those formulas clearly show that TEM waves..."

 

[the_emi_guy]

In a metallic waveguide we know there are no TEM modes.

You cannot extrapolate it to arbitrarily high frequency, relative to size of waveguide, and expect the "no TEM mode" rule to make sense.
For example, consider a waveguide the size of a trashcan and a collimated laser beam shining right down the middle. There is no interaction between waveguide walls and light waves.
On the other hand the core of a fiber optic cable propagates infared light with restricted propagation modes in the same way as large metallic waveguides do for microwaves.
The important distinction is whether or not the electromagnetic wave in question is being *guided* by the waveguide or not.

 

[vanhees71]

Yeah I know that Transverse Electric (TE) and TM modes can exist in these type of waveguides but I was under the impression light we see has to have an Electric and Magnetic component.

Does that mean that we can see EM waves with no M or no E field?

----

Edit: I should have been more clear, I meant "Those formulas clearly show that TEM waves..."

That reveals your misunderstanding! TE or TM just means that the component of the electric or magnetic field, respectively, in direction of the waveguide's axis (usually chosen as the $z$ axis of the coordinate system) is vanishing. Electromagnetic waves always have non-vanishing electric and magnetic fields. That's already clear from a very qualitative perspective on Maxwell's equations: If there is a time-dependent electric field there must be also a magnetic field and vice versa. Only static fields can be solely electric or magnetic.