# The field in a waveguide at exactly the cut-off frequency

Can anyone explain to me qualitatively, what kind of field exists inside a waveguide at EXACTLY the cut-off frequency?

I would be grateful if an explanation was given about waveguides with metalic boundaries and dielectric waveguides (optical fibers).

• I know that below cutoff frequency (for a specific mode) there is attenuation thus no propagation. Above cutoff, there is propagation. The question is simple: What happens EXACTLY at cutoff? Dec 10, 2021 at 18:16

One way of looking at it (not thinking too hard about the differences between TE, TM, and hybrid modes) is to consider the momentum vector, or k-vector, of the mode.

Like any vector, a k-vector can be decomposed as $$\\vec{k}=k_x \hat{x} + k_y \hat{y} + k_z \hat{z}\$$. If the waveguide is aligned in the z-direction, then $$\k_x\$$ and $$\k_y\$$ tell us how much of the wave's momentum is (speaking crudely) bouncing back and forth between the walls of the waveguide, and $$\k_z\$$ tells us how much of the momentum is devoted to propagating the wave along the guide.

The boundary conditions for the (n,m)'th mode to propagate can be expressed as $$k_x=\frac{n\pi}{\ell_x}$$ and $$k_y=\frac{m\pi}{\ell_y}$$

As the waveguide dimensions $$\\ell_x\$$ and $$\\ell_y\$$ shrink, more of the k-vector must be used to fulfill these boundary conditions and less is available for the $$\k_z\$$ component.

When $$\|\vec{k}|^2 = k_x^2 + k_y^2\$$, and therefore $$\k_z=0\$$, we have reached the cut-off condition.

Basically we have a wave that is resonating in the x and y directions (transverse to the guide) and simply a plane wave (no variation) in the z direction.

• Great explanation! Thanks! Does the same applies to optical fibers? Dec 10, 2021 at 18:29
• @MinasMichaelOpethian, essentially yes, but of course the mode structure is different (and there can be many polarizations, not just x and y). The basic result (at cut-off you have no energy transport along the fiber axis) is the same. Dec 10, 2021 at 19:32
• Is there resonance in the xy plane in this case? Dec 10, 2021 at 19:33
• @MinasMichaelOpethian, yes, there would be some kind of resonance in the x-y plane. Dec 10, 2021 at 19:34
• Thanks! Is there any book you recommend that gives nice qualitative explanations like you did? Dec 10, 2021 at 19:37