Terahertz-Range High-Order Cyclotron Harmonic Planar Gyrotrons with Transverse Energy Extraction

Abstract

We develop a multi-mode time-domain model for theoretical description of mode competition in a planar gyrotron with sheet beam of rotating electrons and transverse energy extraction. Within the frame of this model, we demonstrate an improvement of mode selectivity and power increasing in the case of excitation of high-order cyclotron harmonics in terahertz-range planar gyrotrons. The obtained results are confirmed based on direct 3D PIC (particle-in-cell) simulations.

Appendix

Here, we derive the equations for field evolution taking into account the partial penetration of radiation through the collector narrowing. For simplicity, we will further neglect the Ohmic losses as well as the inhomogeneity of the field along the “open” coordinate X in the equation for TE_n mode amplitudes (cf. Eq. 5a):

$$ i\frac{\partial^2{a}_n}{\partial {Z}^2}+{s}_n\frac{\partial {a}_n}{\partial \tau }-i{s}_n{\varDelta}_n-i{\delta}_n(Z){a}_n=0, $$

(A.1)

where the function

$$ {\delta}_n(Z)={\left(\frac{\beta_{\parallel 0}{\lambda}_H}{\pi {V}_{\perp 0}^2}\right)}^2\left({\omega}_n^2(Z)-{\overline{\omega}}_n^2\right)=\left\{\begin{array}{l}0,\kern0.33em Z\in \left(0,{L}_z\right]\\ {}{\overline{\delta}}_n,\kern0.33em Z>{L}_z\end{array}\right., $$

(A.2)

describes the longitudinal profile of the resonator (see Fig. 1b), ω_n(Z) is the TE_n mode cutoff frequency in cross section Z.

The boundary condition a_n(Z = 0) = 0 allows us to present the longitudinal structure of the field as:

$$ {a}_n=\left\{\begin{array}{l}{a}_{I,n}{e}^{i\varOmega \tau}\sin \left({\chi}_IZ\right),\kern0.33em Z\in \left(0,L\right]\\ {}{a}_{II,n}{e}^{i\varOmega \tau -i{\chi}_{II}\left(Z-L\right)},\kern0.33em Z>L\end{array}\right., $$

(A.3)

where Ω is the complex frequency, and

$$ {\chi}_I^2={s}_n\left(\varOmega -{\varDelta}_n\right)\kern0.37em \mathrm{and}\;{\chi}_{II}^2={s}_n\left(\varOmega -{\varDelta}_n\right)-{\delta}_n $$

are the longitudinal wavenumbers inside (zone I in Fig. 1b) and outside (zone II) of the interaction space, respectively. If the planar cavity is weakly perturbed,

$$ {\chi}_I=\frac{m\pi}{L_z}+{\tilde{\chi}}_{nm},\kern0.99em \left|{\tilde{\chi}}_{nm}\right|<<\frac{m\pi}{L_z},\kern1.32em m=1,2,\mathrm{3...} $$

(A.4)

Using the continuity of the transverse electric and magnetic fields in the cross section Z = L_z, we obtain the characteristic equation which determines eigenfrequencies Ω

$$ tg\left({\chi}_IL\right)=i\frac{\chi_I}{\chi_{II}}. $$

(A.5)

Thus, as the first approximation of the perturbation theory, we obtain for ${\tilde{\chi}}_{nm}$:

$$ {\tilde{\chi}}_{nm}=i\frac{m\pi}{L_z^2\sqrt{\chi_I^2-{\overline{\delta}}_n}}\approx i\frac{m\pi}{L_z^2\sqrt{{\left(\frac{m\pi}{L_z}\right)}^2-{\overline{\delta}}_n}}=i\frac{m\pi}{L_z^2\sqrt{D_{nm}}}. $$

(A.6)

When ${\overline{\delta}}_n>{\left( m\pi /{L}_z\right)}^2,\kern0.66em {D}_{nm}={\left( m\pi /{L}_z\right)}^2-{\overline{\delta}}_n<0$, the value of ${\tilde{\chi}}_{nm}$ is real and the corresponding resonator mode with m longitudinal variations is locked by the collector narrowing. In the opposite situation, when D_nm > 0 and the value of ${\tilde{\chi}}_{nm}$ is imaginary (χ_I is a complex value), the effective diffractive losses appear for the resonator mode. Thus, varying the size of the cavity narrowing at the collector side allows for discrimination of modes with D_nm > 0, namely those with large $m>\sqrt{{\overline{\delta}}_n}{L}_z/\pi$.