Abstract
The propagation of slow waves in a dielectric tube surrounded by a long cylindrical metallic waveguide is investigated. The dielectric tube located in a background region of plasma under two different states A and B. In the A-state the dielectric tube hollow filled with the plasma and in the B-state the outer surface of dielectric tube has been covered by the plasma layer. There are two relativistic electron beams with opposite velocities injected in the waveguide as the energy sources. Using the fluid theory for the plasmas, the Cherenkov instability in the mentioned waveguide will be analyzed. The dispersion relations of E-mode waves for the states A, B have been obtained. The time growth rate of surface waves are compared with each other for two cases A and B. The effect of plasma region on time growth rate of the waves, will be investigated. In all cases it will be shown, while an electron beam is responsible for instability, another electron beam plays a stabilizing role.
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Appendix
Appendix
1.1 Derivation of the perturbed relativistic equation of the motion, and the continuity equation for the beam electrons presented in Sect. 2
The equations to describe the system are the equation of motion and the continuity equation
where n and V are the density and the velocity of particles (electrons of plasma and beam), and all other terms have their usual meaning. The interaction of the electromagnetic waves with the plasma will disturb the equilibrium density and velocity of the particles and electromagnetic fields. These disturbtions will be analyzed by considering the perturbation variables n, V, E and B. We can define them as follows:
where in the system of equations, the small deviations from the equilibrium state is considered as (\(\delta n\ll n_{0}\), \(\mid \overrightarrow{\delta V}\mid \ll \mid \overrightarrow{ V}_{0} \mid\), \(\mid \overrightarrow{\delta E}\mid \ll \mid \overrightarrow{ E}_{0} \mid\), \(\mid \overrightarrow{\delta B}\mid \ll \mid \overrightarrow{ B}_{0} \mid\)). Also, all of the perturbations are described as \(\delta \psi = \delta \psi _{0}e^{-i(\omega t-\beta z)}\). Using (24)-(26), we can derive the expressions for Eqs. (1) and (2). Form Eqs.(1) and (2), the perturbed density and velocity of electron beam and plasma layer will have the form:
Now, we can derive the expressions for the charge and current density. The charge and current density \(J,\rho\) are defined through the electron plasma and beam quantities as:
Eqs. (26)–(29) give us Eqs. (3) and (4).
1.2 Derivation of the solutions to the wave equations presented in Sect. 2
Using Eqs. (3)–(8), the solutions of the wave equation for all of the regions of given cylinder waveguide can be presented. For example, the wave equation in the core region with plasma column in the model A is
Substituting Eqs. (3)-(4) and using linear approximation of monochromatic plane waves for all of the perturbed quantities, one can derive
By introducing the parameter \(\Gamma ^{2}_{b+}(1,\omega _{b_{+}},\omega _{p})\) (like Eq. (8)), the z-component of the perturbed electric field \(\delta E_{z}\) is obtained as Eq. (8). It is obvious that the solutions of Eq. (8) are the modified Bessel functions of the first and second kinds. Regarding the modified Bessel function of the second kind has a singularity at the origin \((r = 0)\), the modified Bessel function of the first kind is acceptable for this region as Eq. (10). The solving method in other regions is the same.
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Hajijamali-Arani, Z., Jazi, B. A description on plasma background effect in growth rate of THz waves in a metallic cylindrical waveguide, including a dielectric tube and two current sources. Indian J Phys 92, 1307–1318 (2018). https://doi.org/10.1007/s12648-018-1198-0
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DOI: https://doi.org/10.1007/s12648-018-1198-0