VOLUME 56, NUMBER 25 PHYSICAL REVIEW LETTERS 23 JUNE 1986 Linear and Nonlinear Theory of Cherenkov Maser Operation in the Intense Relativistic Beam Regime Don S. Lemons and Lester E. Thode Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received 27 January 1986) The linear dispersion relation for axisymmetric TM modes in a cylindrical waveguide lined with a dielectric material and enclosing a thin annular electron beam is derived and solved. Approximate analytic solutions are obtained for both weak and moderate beam regimes. A model of nonlinear saturation is developed. This model, combined with the linear theory in the moderate beam re- gime, defines the most efficient and compact Cherenkov maser. PACS numbers: 42.52.+x, 52.35.Mw, 52.40.Db, 85.10.Hy There have been a number of investigations of the so-called Cherenkov maser—an electron beam in- teracting with a dielectric-lined waveguide. Previous linear analysis confines itself to the '*weak beam re- gime,"*"^ solid beams,^*^ plane geometry/ or the nu- merical solution of various dispersion relations that do not delineate the various aspects of the coupling. Nonlinear analyses and simulations^'^'^'^^ have limited the problem to interactions that are nonrelativistic in the wave frame. A number of different experiments have also been carried out.*^'^'^^'^^ Independently, high-brightness, intense, relativistic electron beams have been generated by use of a foilless diode.^^'^"^ Although developed for an inertial-confinement fusion concept that used the relativistic two-stream instabili- ty,^^ this class of relativistic electron beam might be applicable to high-power Cherenkov masers. Thus, we present a linear and nonlinear theory of the Cherenkov maser in a newly investigated regime associated with an annular, high-brightness, intense, relativistic elec- tron beam. The analysis suggests that a high-power Cherenkov maser with 30% efficiency may be possible with this type of electron beam. First, we obtain the linear dispersion relation for ax- isymmetric modes on a thin annular beam in a dielectric-line cylindrical waveguide. The present der- viation is similar to previously published ones^ and will only be outlined. Since axisymmetric TM modes decouple from TE modes, we need only consider the wave equation for the TM modes. This wave equation is solved in three regions, the first between the center of the waveguide and the annular beam, the second between the annular beam and the dielectric, and the third inside the dielectric. In general, each solution is expressed in terms of two independent eigenfunctions yielding a total of six constants to be determined by four boundary and two jump conditions. The boun- dary conditions are as follows: The longitudinal elec- tric field E^ is finite at the origin, E^ vanishes at the conducting waveguide wall, and both E^ and the radial electric displacement eEj. are continuous across the vacuum-dielectric interface. Here e is the dielectric constant. In the thin-beam approximation, the two jump conditions [E,] = 0 and [rEr]^2eikEJJ{mv^yl)U- kv,y (1) (2) are derived respectively from the Faraday and Gauss laws with a linearized cold-fluid description of the beam. In Eqs. (1) and (2) the square brackets indicate the enclosed quantity evaluated at the upper side of the beam at r = r^ minus the quantity evaluated at the lower side of the beam at r = rf. Furthermore 4, m, e, v/,, and y^ are respectively the beam current, elec- tron rest mass, electron charge, electron velocity, and Lorentz factor. For axisymmetric vacuum TM modes E^ and £j are related by £,= (//c64/9r)(coVc2-^2)-i We have assumed a dependence of exp{/(A:2-a>r)} for the fields. The resulting dispersion relation is where Z)i = e^[yo(pr^)Fi(pr^)-/i(pr^)ro(p^v.)], £>2 = pl^o(pA->.) FoCpr^)-/oCp^rf) J'oCp^w)]. Qi = Ji(ir,)Yoi^r,)~U^r,)Y,{^r,). Qj-Joi^r,)Y^{^r,) - J^^rt)Y^{^r,), ^'^={o?/c^~k^), p^==(a>^€/c^-/r^), and i„ and Y^ are Bessel functions of the first kind and nih order. Also ^l-^kc^/ril^bylh. where /o = ^/ mr. Here r^ stands for the conducting wall radius and r^ for the beam radius. The dielectric extends from an inner radius r^ to r^,. Equation (3) may also be recovered from Eq. (17) of Ref. 1 2684 © 1986 The American Physical Society