Something has gone wrong with your simulation. Below cutoff the wave is reflected. Energy is conserved. The evanescent wave has inductive or capacitive energy that builds up and releases each half-cycle, but does not propagate or absorb energy. There will also be some resistive losses, except in PEC or PMC.

If the cut-off section is short there can be tunnelling and the wave can propagate on the other side.

As Malcolm said, it is accumulated near the entrance/feed point. Just for clarification, it can be any reactive accumulation depending on the structure, and if it is a hard cut-off, the distribution will be decaying exactly by exponential law... so it really can leak at the other side if the wg section is comparably short. in general, the energy is reflected back, minus negligible conductive loss at that decay range. On practice, cutt-offs are not hard, so there is a small β along with large α, making the exponent distribution oscillating, in some cases with energy decaying faster than predicted, then suddenly appearing at farther range. When you have troubles with numerical simulation, it may be the case; you should enlarge your model.

In order to simplify, you can consider a lossless closed waveguide. In this case, an EM simulator give a return loss (S11) = 0 dB for frequencies under cutoff. This result shows that the energy is entirely reflected toward the source.

Hello,

All waveguides have a cut-off frequency. Below this cut-off frequency the waveguide is unable to support power transfer along its length. Waveguides will only carry or propagate signals above a certain frequency, known as the cut-off frequency, and the cut-off frequency of the waveguide depends upon its dimensions. So, in that way, S11 is approximately equal to zero, and having imaginary value, so it will attenuated. No EM wave will be propagated inside the waveguide.

Thanks,

Be careful with time domain e.m. simulator. FDTD method, for example, may give wrong results if you stop simulation too early, especially with frequencies under cut-off as the energy decay inside the structure is very slow.

Marcello, that's a good point. FDTD simulations do not work well with low-loss models. If the energy is not lost and does not go to load, it can oscillate in reactive components almost forever, accumulating numeric errors. even when such errors are not a problem, there are still difficulties in extracting cavity/under-cut-off modal data if the excitation pulse did not finish its life in the load (see harminv, a "harmonic inversion" technique).

Your question is very interesting. I agree with Dr. Jinfeng Li's answer

Waveguides are metal tubes functioning as “conduits” for carrying electromagnetic waves The wave travel along a standard, two-conductor transmission line is of the TEM (Transverse Electric and Magnetic) mode, where both fields are oriented perpendicular to the direction of travel.

Below the cut off frequency the electromagnetic waves are not capable to propagate in the wave guide. So they will be reflected at the port of the wave guide. The wave guide works as a high pass filter. The filter here is not a dissipating one but it is a reflection type filter.

I would like to ask you what is the characteristic impedance of the wave guide that you considered to calculate the reflection coefficient?

The answer of this question may lead you to trouble shoot the error.

Best wishes

Below cut off the wave will not enter.

Please read my book

The American University Laboratories For Electrical Engineering Part 2

I hope you find a good theory in it.

Dear Y. Song

I would suggest the following research article in this regard.

https://www.electronics-notes.com/articles/antennas-propagation/rf-feeders-transmission-lines/waveguide-cutoff-frequency.php

Thanks Tariq, article very useful

I don't know why Dr. Zekry's reply got so much likes, but I want to warn everyone supporting that idea that a large class of devices, called "metamaterials", including antennas, filters and splitters, is basically based on propagation below cutoff. It is an evanescent field leaking that can be easily turned into arbitrary propagation mode, by compensating the reactance at finite distance. If you don't take that into account, you can drown in countless EMC problems in your design.

Hi, Andrey Porokhnyuk

What was it about Dr Zekry's reply you didn't like - I haven't been able to find any mention of metamaterials?

Hi, Malcolm. Here, I am speaking about "Below the cut off frequency the electromagnetic waves are not capable to propagate in the wave guide. So they will be reflected at the port of the wave guide. "

Which is not exactly the truth. It is a lumped element model, which has its spatial limits in real application. Moreover, in reality, cut-offs are not strict, so the propagation constant may be complex with dominant imaginary α decay, but still with considerable real part β

Hi Andrey Porokhnyuk

There is only a real part if the guide becomes big enough to propagate at some point further on, or if there is some structure at a similar distance, like a post or something that can support propagating fields - as in an evanescent mode filter. There will also be a real part associated with loss in the walls of the evanescent waveguide. Otherwise there is no real part, and the cut-off guide looks like a pure inductor (or perhaps capacitor - depending on which mode is used?).

With good conductors and a cutoff waveguide about 10 times as long as it is wide, the real part is very small and can be ignored for most practical purposes.

About the only time propagation over a significant length of cut-off waveguide would be significant would be in a cutoff attenuator, where a probe could be inserted a variable distance along the cutoff section to get well-controlled variable coupling with very high values of attenuation. Then the size of the real part along the whole of the cutoff guide would vary with the position of the probe.

with a ferrite I can easily synthesize almost any complex propagation constant. And in real life there is always some dielectric loss, or surface imperfection. There is simply no such thing as strict cutoff. And when you put something closer than a few decay lengths to the cutoff section... that will be the answer, "where power goes".

It should be recalled that below cutoff the field does not stop abruptly but still extents a bit (say up to 1 free space wavelength for 1/e decay) into the waveguide below cutoff region.(can be visualized via quasi-static approximation).There is also a considerable number of papers claiming superluminal propagation (in the evanescent field region) for waveguide modes below cutoff..but this discussion would open another "can of worms" .Anyway the approach violates the (what I call) "Jackson Caveat" (Jackson textbook on EM) which states that the normal definition of group velocity is not applicable in regions of strong dispersion (e.g. waveguide below cutoff) other wise one can obtain in the frame of the same logic, negative group delays (signals arrived before it was sent)

Recent papers about quantum tunnelling suggest that takes finite time, so evanescent fields might too, but I'm not sure. Evanescent and tunnelling may be exact equivalents.

As soon as any power is coupled, as when the waveguide gets bigger again, there is a reflected evanescent wave and it gets more complicated.

I tried measuring the delay using a radar pulse in a waveguide, and reducing the carrier frequency. It got very slow, with large delay just above cut-off. Below cut-off it got a lot faster, but my guide wasn't long enough to tell if it was speed of light or instant, especially with the very dispersive behaviour which destroyed the pulse shape!

Dear Fritz, that thing you are talking about is called metamaterials, and prior that, the same thing was called periodic structures, supporting backward-wave propagation. It even exists in physical media, like ferrites, without any periodicity or bandgaps. Delays are not negative, but the phase flow is. If you solve the wave equation carefully, you will see how the dispersion branch flips on the other side of negative β, which means exactly that. Sometimes it is called a negative refraction, or even negative refraction coefficient is stated (which is actually not negative... but the engineering slang is a persistent thing).

And yes, it is very easy when you treat the wave number not as β, but as complex γ=α+jβ, which puts everything on its place.

Dear Andrey

thanks for your notes. I have this structure, which is a rectangular waveguide with some metallic post inserted into the top and bottom (with screws) of the waveguide. so my cut of frequency is 8.3 GHz but when I simulate this structure I get dispersion diagram which is below 5 GHz.My \Beta is still positive ? Andrey Porokhnyuk

What you have can be thought of as a 1D metamaterial or as an evanescent mode filter, in which the resonators are the posts in their short section of waveguide, and the coupling is through the evanescent modes between.

http://www.g4jnt.com/EVANFILT.pdf

Conceptually there are two type of losses, the dissipation losses in resistive like loads and reflection losses.

In reflection losses there is no energy dissipation and conversion into other forms.

The power is reflected to the source again.

Best wishes

Nasr Alkhafaji

It is not absorbed in any significant amount - it is mostly reflected. An experiment with a reflectometer or Vector Network Analyser will show close to 100% reflection. Waveguide does get more lossy close to cut-off, but it would need to be a very gradual taper for the part close to cut-off to be long enough for the losses to be significant if the metal is a good conductor.