1. RF Cavities & Waveguides1 1
RF Cavities & Waveguides
Jeffrey Eldred
Classical Mechanics and Electromagnetism
June 2018 USPAS at MSU 1 1 1 1 1 1

2. 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
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3. Standing Waves
Consider two counterpropagating waves of the same frequency: 3
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4. Standing Waves
Consider two counterpropagating waves of the same frequency: 4
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5. 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
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6. Separation of Variables Waveguides or Cavities6 6
Separation of Variables
Waveguides or Cavities 6
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7. 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
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8. TE & TM Waveguide Modes TE Modes TM Modes 8 8 8 8 8
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9. 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
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10. Separation of Variables
Dividing the whole expression by Ψ:
For this expression to be true, each term must be a constant. 10
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11. Rectangular Transverse Modes
For rectangular geometry:
For a TM wave the solutions are:
For a TE wave the solutions are: 11
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12. Rectangular Transverse Modes
Pozar 12
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13. 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
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14. Cylindrical Transverse Modes
TM:
TE:
Pozar 14
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15. 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
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16. Waveguides In the plane wave, we had:
But that’s only for a plane wave. Generally we have:
Phase-velocity:
Group velocity: 16
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17. RF Cavities Boundary Conditions: For a TM waves and TE waves we have:
We are confined to discrete frequencies: 17
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18. 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
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19. Transverse Fields Transverse Magnetic (TM) Modes:
Transverse Electric (TE) Modes:
Where Z is the characteristic impedance. 19
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20. 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
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21. 21 21
Cavity Concepts 21
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22. Bead Pull Measurement Oak Ridge 22 22 22 22 22
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23. Slater’s Formula 23
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24. 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
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25. 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
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26. 26 26
Reflection of EM Waves 26
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27. Reflection of EM Waves Incident, Reflected, Transmitted:
Boundary Conditions:
Solution: 27
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