RF Cavities & Waveguides 1 1 RF Cavities & Waveguides Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU 1 1 1 1 1 1

Maxwell’s Equations in a Vacuum d'Alembertian: Full solution is a plane wave: E1 and E2 are complex, the relative phase determines the polarization. 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 2 2 2 2

Standing Waves Consider two counterpropagating waves of the same frequency: 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 3 3 3 3

Standing Waves Consider two counterpropagating waves of the same frequency: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 4 4 4 4

RF Cavities & Waveguides EM waves reflect off of conductive surfaces. The geometry of those conductive surface determine the frequencies at which the EM resonate between surface. RF cavities confine the EM waves in 3D, and there will only be discrete eigenfrequencies. Generally one is useful the rest are noise. Waveguides confine the EM waves in 2D and transmit EM waves. Each mode of the waveguide has a cut-off frequency. 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 5 5 5 5

Separation of Variables Waveguides or Cavities 6 6 Separation of Variables Waveguides or Cavities 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 6 6 6 6 6 6

Boundary Conditions At the surface of a perfect conductor. Since the boundary conditions on Ez and Bz will be different, the mode decomposition will in general will be different. The wave modes can be cleanly classified as Transverse Magnetic (TM), Longitudinal Electric. Transverse Electric (TE), Longitudinal Magnetic. 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 7 7 7 7

TE & TM Waveguide Modes TE Modes TM Modes 8 8 8 8 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 8 8 8 8

Separation of Variables Where Ψ is Ez for TM or Hz for TE, we can write the Helmholtz Eq: Or we can write this in cylindrical coordinates and obtain: The trick to solving this is separation of variables. Look for solutions for Ψ in the form: Then we can write Ψ as a linear superposition of all such solutions. 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 9 9 9 9

Separation of Variables Dividing the whole expression by Ψ: For this expression to be true, each term must be a constant. 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 10 10 10 10

Rectangular Transverse Modes For rectangular geometry: For a TM wave the solutions are: For a TE wave the solutions are: 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 11 11 11 11

Rectangular Transverse Modes Pozar 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 12 12 12 12

Cylindrical Transverse Modes For cylindrical geometry: The solutions are: Where xmn is the nth zero of the mth Bessel function Jm(x). Where x’mn is the nth zero of the mth Bessel function J’m(x). TM: TE: 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 13 13 13 13

Cylindrical Transverse Modes TM: TE: Pozar 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 14 14 14 14

Waveguides Solve the expression for time: For a waveguide, this is just an oscillation with wavenumber k γ is discrete but k is continuous. So for each mode there is a cutoff frequency at k = 0 and there is a fundamental cutoff frequency. 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 15 15 15 15

Waveguides In the plane wave, we had: But that’s only for a plane wave. Generally we have: Phase-velocity: Group velocity: 16 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 16 16 16 16

RF Cavities Boundary Conditions: For a TM waves and TE waves we have: We are confined to discrete frequencies: 17 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 17 17 17 17

Eigenmode Basis The solutions to the Helmholtz equation are eigenmodes, each with its own eigenfrequency. Any collection of E & B fields inside the cavity could be described by some linear combination of eigenmodes. There would be many functional basis we could use to describe the E&B fields, but eigenmodes are special because they are coherent. An arbitrary excitation of many cavity modes will rapidly decohere. However, the modes themselves will only decay very slowly. 18 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 18 18 18 18

Transverse Fields Transverse Magnetic (TM) Modes: Transverse Electric (TE) Modes: Where Z is the characteristic impedance. 19 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 19 19 19 19

Laplace’s Equation We showed how separation of variables works for Helmholtz Eq: But we can use a similar technique to solve Laplace’s Equation: And impose boundary conditions of the form: Which would allow us to find the electric field from voltage distributions on a simple geometry. 20 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 20 20 20 20

21 21 Cavity Concepts 21 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 21 21 21 21 21 21

Bead Pull Measurement Oak Ridge 22 22 22 22 22 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 22 22 22 22

Slater’s Formula 23 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 23 23 23 23

Higher Order Mode (HOM) Chokes Generally only the lowest-order mode, The “fundamental mode” is used to accelerate the beam. All the higher order modes (HOMs) are undesirable noise. So keep the HOMs away from the beam and dissipate them with chokes. Cavities can also be designed so that HOMs occur at higher frequencies. 24 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 24 24 24 24

Other Types of Cavities Multi-mode cavities deliberately design the HOMs to be some harmonic of the fundamental frequency and to have beneficial effects on the beam. Transverse deflecting cavities are used to kick the beam as a whole or to introduce a longitudinal-dependent position offset. Example of both concepts: Huang et al. JLAB 25 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 25 25 25 25

26 26 Reflection of EM Waves 26 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 26 26 26 26 26 26

Reflection of EM Waves Incident, Reflected, Transmitted: Boundary Conditions: Solution: 27 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/9/2018 27 27 27 27