Dispersion relation of the electromagnetic waves propagating through a helix inserted into a magnetron-type conducting cylinder

The dispersion relation of the electromagnetic waves propagating through a helix is obtained for the helix inserted into a magnetron-type cylindrical conductor. The resultant dispersion relation consists of a combination of helix and magnetron dispersion relations. In special cases, they decouple from each other and recover the previous results. For example, in the case when the magnetron vane structures disappear, the dispersion relation recovers the typical helix dispersion relation. On the other hand, the resultant dispersion relation recovers the magnetron-transverse electric mode when the helix pitch angle φ approaches π/2.