Appendix A Gyrotron Theory A.1 Calculation Procedure The gyrotron is a maser device for which the underlying mechanism is normally understood within the frame work of quantum mechanics as based on transitions between Landau levels rather than using transitions between atomic levels. From this perspective, gyrotrons are more specifically cyclotron masers. However, describing the gyrotron mechanism quantum mechanically is not practical. Instead, classical considerations using Maxwell’s equations and the Lorentz force are usually used [1, 2] and give a satisfactory understanding of gyrotron behavior. First, we introduce electromagnetic fields into the RF cavity in the absence of an electron beam. Electron- beam handling is then discussed. Next, the interactions between the millimeter waves and the electron beam are calculated. This formalism is called the cold cavity method, in which perturbations from the electrons to the profile of the radiation field is ignored. Another formalism, called the self consistent method, enables the electron motion and RF behavior to be solved simultaneously. The cold cavity formalism is a good approximation when the quality factor of the RF cavity is sufficiently high. A.2 Theory of the RF Cavity The RF cavity is an open-ended cylindrical resonator that has a characteristic angular frequency of resonance, which we denote by ωr. The resonator can be thought of as an irregular waveguide because the radius (R) of the helical motion of the electrons varies slowly along the z-axis as plotted in Fig. A.1. Electrons from the MIG are introduced, where the resonator is cut off (left-hand side of figure), preventing RF fields from leaking and damaging the MIG. The intermediate region constitutes a flat waveguide of radius R2 and length L2. The output section (right-hand side of figure) is open. © Springer Japan 2015 97 A. Miyazaki, Direct Measurement of the Hyperfine Structure Interval of Positronium Using High-Power Millimeter Wave Technology, Springer Theses, DOI 10.1007/978-4-431-55606-0 98 Appendix A: Gyrotron Theory Fig. A.1 Cross-sectional geometry of the RF cavity used in the gyrotron FU CW GI The resonance mode of the RF cavity is a solution to Maxwell’s equations in a cavity wall. We shall use cylindrical coordinates, (r,φ,z) in accordance with the symmetry of the cavity. Only the transverse electric (TE) modes are considered because gyrating electrons couple to the transverse electric fields and not with the magnetic fields. Upon introducing an azimuthal index (m = 0, ±1, ±2, ±3, ···), a radial index (n = 1, 2, 3, ···), and an axial index (l = 1, 2, 3, ···), the eigenmodes of this cavity (TE resonance modes) are products of a transverse vector function, −→ mnl E (r,φ) and an axial scalar function, (z). One can approximate the transverse −→ electric field ( E (r,φ))bytheTEmn modes propagating through a circular waveguide, expressing them as: R j E (r,φ)= E m J mn r sin(mφ), (A.1) r 0 r m R jmn Eφ(r,φ)= E J r cos(mφ), (A.2) 0 m R Ez(r,φ)= 0, (A.3) where E0 is the normalized strength of the electric field, Jm(x) the m-th Bessel ( ) ( ) = function of the 1st kind, Jm x its 1st derivative, and jmn is the n-th root of Jm x 0. As the taper angles of the cavity are small, the changes in R(z) can be ignored and −→ mode mixing with other TEmn s is also negligible. FigureA.2 gives E (r,φ)for the TE52 used in this experiment. Note that there are two degenerated states according to the sign of the azimuthal index (m), except for when m = 0, which corresponds to a phase rotation around the z-axis. When gyrating electrons are present, this degeneracy is removed and the electron motion becomes either co-rotating or counter-rotating with respect to the magnetic field. Appendix A: Gyrotron Theory 99 Fig. A.2 Transverse 3 structure of the electric field in the RF cavity (TE52 mode) 2 1 0 y [mm] -1 -2 -3 -3 -2 -1 0 1 2 3 x [mm] The axial function ((z)) obeys the equation describing a nonuniform string: d2(z) + ζ 2 (z)(z) = , 2 mn 0 (A.4) dz ω 2 j 2 ζ (z) = − mn , (A.5) mn c R(z) where ω = ω 1 + i is the complex RF frequency and Q the diffraction r 2QD D quality factor. Corrections for the open-endedness of the cavity were calculated for the cutoff and output sections to estimate ωr and QD simultaneously. Equation (A.4) can be solved numerically under the following boundary conditions: d (0) = iζ (A.6) dz mn d (z ) =−iζ , (A.7) dz out mn where zout is the end-point of the output taper (zout = L1 + L2 + L3). Equation (A.4) can be solved using Eq. (A.6) as the initial condition, for example, using the Runge-Kutta formula. Next, the two variables (ωr, QD) are varied until the sec- ond boundary condition, Eq. (A.7), is satisfied. FigureA.3 shows the result of this calculation for the RF cavity used in this experiment. The three different curves cor- respond to different axial indices (l = 1, 2, 3). The solutions are listed in Table A.1. The quality factor from the Ohmic loss (Q) and the combined Q are also shown. 100 Appendix A: Gyrotron Theory 120 100 80 60 (z) [a.u.] Ψ 40 20 0 0 5 10 15 20 25 z [mm] Fig. A.3 Axial structure of the electric field in the RF cavity. The solid, dotted,anddashed lines show the TE521,TE522 and TE523 modes, respectively Table A.1 Resonance frequency and QD of the three modes shown in Fig. A.3 Mode Frequency (GHz) QD Q Q TE521 203.01 3200 10600 2500 TE522 203.59 810 10600 750 TE523 204.56 380 10600 370 The cavity parameters are (L1, L2, L3) = (5, 14, 5mm) and (R1, R2, R3) = (2.345, 2.475, 2.735 mm) The Ohmic quality factor is given by: R2 m Q = − , δ 1 (A.8) jmn where δ = 0.18 µm is the skin depth of copper at 200 GHz. Under normal operations of the gyrotron (gyromonotron), the TE521 mode (the solid line in Fig. A.3) is excited. Interactions with the backward-wave are needed for the electrons to couple to higher axial modes. This device is called a gyro-BWO and has the potential to continuously tune the output frequency. The output power of this device is currently less than 100 W, which is insufficient to measure the Ps-HFS. A proposal on how to achieve higher power is suggested in the future prospects section (Sect. 4.3). A.3 Theory of the Electron Beam An electron beam should be properly controlled and introduced into the correct position in the RF cavity to obtain maximum coupling between electrons and RF Appendix A: Gyrotron Theory 101 fields. A sophisticated method to control the electrons has been developed in gyrotron physics. The simplest approximation is described in this section. Electrons are emitted from an emitter ring at the cathode and are accelerated between the 1st and 2nd anodes (Fig.2.2). Guided by the magnetic field, they enter the RF cavity and gyrate where the magnetic field strength is maximal at about 7.5 T (Fig. A.4). Finally, the electron beam is dumped by a beam collector (Fig. 2.2). FigureA.5 shows the electron trajectories simulated using CST PARTICLE STU- DIO [3]. The radius of the electron beam is compressed by the strong magnetic field in the cavity. Depending on the beam radius (Rb), electrons maximally couple to one specific mode (TEmn1) in the RF cavity, as described in Sect. A.2. This coupling strength is determined by the function: 8 7 6 5 4 Bz [ T ] 3 2 1 0 0 200 400 600 800 1000 z [mm] Fig. A.4 Simulated magnetic field (Bz) along the z-axis (simulated using CST PARTICLE STU- DIO). The emitter ring is at z =0 mm and the RF cavity is at 650mm Fig. A.5 Simulated electron trajectories from the MIG (simulated using CST PARTICLE STUDIO) 102 Appendix A: Gyrotron Theory Fig. A.6 Coupling factor of the TE52 mode. The solid 0.025 line represents the counter-rotating mode and 0.02 the dashed line the co-rotating mode 0.015 G 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rb/R 2 J ∓ j Rb/R G = m 1 mn , (A.9) mn ( 2 − 2) 2 ( 2 ) jmn m Jm jmn where the sign of m ∓1 signifies either co-rotating or counter-rotating wave propaga- tion relative to the direction of the electron gyration. FigureA.6 shows the calculated G52. It is clear that the degeneracy of the co- and counter-rotating modes is removed. When the gyrotron oscillator is operated in the counter-rotating TE52 mode, Rb should be controlled so that the wave covers 0.7 of the cavity radius R2. Depending on magnetic field strength, the beam radius (Rb) is compressed, as demonstrated in Fig.A.5. This adiabatic compression is expressed as: Bc Rb(z) = Rc, (A.10) Bz(z) where Bc ∼ 0.1 T is the z−component of the magnetic field at the emitter ring at the cathode, and Rc ∼ 14 mm is the ring radius (Fig. 2.2). The factor b = Bz(z)/Bc is called the compression ratio.