In reality, not every choice of mode numbers leads to a real field. The most obvious example of this is when 
. This choice means that 
, i.e. no field anywhere in the guide! This mode therefore can't actually exist — or at least, it can't every carry any power! Large values of the mode numbers can also lead to a non-existing mode. To see why, consider again expressions 10.3 & 10.4. The exponential term, 
, indicates how the field phase varies along the guide (and with the passage of time). In order for the mode to be able to Propagate along the guide we must ensure that the Propagation Constant, 
, is real. This ensures that the exponential term represents a sinusoidal phase variation with the position along the guide, z. An imaginary value for 
would mean that the field changes exponentially with z. Any attempt to launch such a field along the guide produces a field pattern which rapidly (exponentially, in fact!) declines along the guide. Such a mode will not therefore propagate and carry power along the guide.
Combining expressions 10.6 & 10.7 and rearranging we can say that the requirement for a real propagation constant means that the signal frequency must be such that
The lowest frequency value which will satisfy this expression for any choice of mode numbers is called the mode's cut-off frequency,
This result means that only when the signal frequency is greater than or equal to 
will it be possible for the m,n mode to propagate along the guide. The mode won't propagate when we try to make it carry power at a frequency ‘below cut-off’. This means that the total field in the guide is always limited to being a linear superposition of those modes which can actually propagate at the signal frequency.
Standard rectangular microwave guide has a 2:1 aspect ratio, i.e. 
. Since 
we can say that the cut-off wavelengths of the first few modes in such a guide will be
where 
is the free-space wavelength of the cut-off frequency of the mn'th mode. Table 10.1 shows the relative cut-off wavelengths of the first few modes in a standard rectangular waveguide.
Table 10.1
Cut-off wavelengths of TEmn modes in standard
waveguide.
| 
| 
| 
| 
|
| — | 2 | 1 | 0·6666 |
 | 1
| 0·8944 | 0·7071 | 0·5477 |
 | 0·5
| 0·4850 | 0·4000 | 0·3535 |
| 0·3333 | 0·3288 | 0·3162 | 0·2981
|
This table reveals two important properties of metallic ‘pipe’ waveguides. Firstly, the lowest cut-off wavelength is 
. No modes exist for longer wavelengths, hence this is the wavelength of the lowest signal frequency which a given size of guide will carry. Lower frequency signals can't propagate along the pipe. In between the wavelengths 
and 
only one mode, the TE01 mode, can propagate. (A similar analysis of the TM-mode family confirms that none of these can propagate for wavelengths below 
.) In this region the guide's behaviour is said to be single mode.
Single mode operation is important for two related reasons. Firstly, in this wavelength/frequency range we can know that the field distribution in the guide can only have the form of the TE01 mode. As a result we can say that, anywhere along the guide, the field will be
The mode with the lowest mode-numbers which propagates is conventionally called the Fundamental Mode.
Secondly, the propagation constant, 
, determines how quickly the field phase varies along the guide. For this reason it is sometimes called the Phase rate or Phase Velocity. Looking at equations 10.6 & 10.7 we can see that the value of 
depends upon the mode numbers. This means that different modes propagate along the guide with different velocities. A Multi-Mode field (one composed of a superposition of various modes) will therefore produce a total field pattern which varies along the guide as the components move in and out of phase. Some modes will also carry power along the guide quicker than others. This tends to ‘smear out’ any information-carrying modulations of the field. Multi-mode fields are therefore usually undesirable when we want to send signals along a guide. Using the guide in the single mode range allows us to avoid these problems.
Content and pages maintained by: Jim Lesurf (jcgl@st-and.ac.uk)
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.