I have calculated and plotted the dispersion relation for a cylindrical wave-guide partially loaded with an artificial dielectric. I now need to calculate and plot my cyclotron electron beam / wave-line on my dispersion plot. Can anyone explain how this is done?
I believe that your electron beam line is a straight line from the origin with a slope equal to the mean electron velocity. That is certainly what it is for magnetrons.
Interaction occurs where the beam line crosses or touches the dispersion curve. At speeds up to the lowest interaction speed there is no net exchange of energy between the beam and the rf field so the mean velocity can be calculated without including rf interaction.
Malcolm White Thanks for your answer. I know this is true for some electron beams. However, in this case the beam is annular and propagates in much the same way as those found in a Gyrotron. The beam-mode dispersion relation I need to calculate is essentially the fast cyclotron wave on a beam of electrons in motion transverse to a DC magnetic field. From the literature the gradient appears to be the beams propagation constant multiplied by the axial velocity. The beam line is also shifted up (in y) by an amount determined by the relativistic angular cyclotron frequency.
Hi Simon Foulkes
Sorry that I am not up to speed on the relativistic beams you are working with.
However - look out for the units used in your curves. Wavenumber times velocity is radians per second, which implies a rotation rate. Presumably the wavenumber gives the rotation of the mode as it progresses along the axis.
You need to make sure that your dispersion diagram has the correct things plotted so that a slope on it is in radians per second - perhaps the beta axis needs to be in radians (of mode pattern) per radian (of physical rotation).
I have a gut feeling that an upward shift (up the frequency axis based on the cyclotron frequency) could make sense, but no help to offer!
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