**3.1 Operation at harmonics of cyclotron frequency**

With the proper shape of the RF-field, it is possible to excite harmonic mode of interactions with the electrons [1, 2, 8, 13–15]. As shown in **Figure 7**, the EM-wave field oscillates at a frequency twice the cyclotron frequency [2], i.e., ω = 2ωc. The direction of the field reverses in the center of the electron orbit. Thus, an electron that is initially decelerated by the field is moving transverse to the field when the field reverses, and so does not have its orbital energy changed. By the time the field reverses again, the electron has moved 900 around its orbit and is again in a decelerating field. Thus, during each full orbital motion of electron, the RF-field goes through two complete cycles. Hence, for harmonic mode of operation, for a given operating frequency, the cyclotron frequency is half the value used in fundamental mode of operation. As a result, the magnetic field is reduced by a factor of two. Operation at frequencies higher than the second harmonic are also being examined [15], but the intensity of the interaction is reduced, making the efficiency of gyrotron low.

For harmonic mode operation, the frequency of operation of the gyrotron is approximately given by

$$
\rho \cong s \left(\frac{\varepsilon}{\gamma m\_0}\right) B\_0 \tag{5}
$$

**Figure 7.**

*Harmonic interaction of an electron and a field varying at twice the cyclotron frequency. (a) At an arbitrary time T, (b) At half RF cycle after T.*

*Gyrotron: The Most Suitable Millimeter-Wave Source for Heating of Plasma in Tokamak DOI: http://dx.doi.org/10.5772/intechopen.98857*

where, s is an integer, representing the harmonic number. Value of s equals to 2 corresponds to second harmonic operation. It signifies that the electromagneticwave frequency of the gyrotron is chosen to be twice the cyclotron frequency. Harmonic operation reduces the magnetic field requirements by factor of s (*B0/s*). For gyrotrons operating at W-band or above, magnetic field requirement is very high (beyond 3 Tesla), it's not possible to obtain such magnetic field from a normal solenoid magnet. This necessitates the use of superconducting-magnets. Harmonic operation is a suitable choice for such class of gyrotrons, as second harmonic operation reduces the magnetic field requirement to half. Hence, such magnets can be built with non-superconducting solenoid coils. However, the harmonic operation reduces the efficiency of the gyrotron.
