Superconducting Radio Frequency - Wakefields and Higher Order Modes (HOMs)

Wakefields and Higher Order Modes (HOMs)

One of the main reasons for using SRF cavities in particle accelerators is that their large apertures result in low beam impedance and higher thresholds of deleterious beam instabilities. As a charged particle beam passes through a cavity, its electromagnetic radiation field is perturbed by the sudden increase of the conducting wall diameter in the transition from the small-diameter beampipe to the large hollow RF cavity. A portion of the particle's radiation field is then "clipped off" upon re-entrance into the beampipe and left behind as wakefields in the cavity. The wakefields are simply superimposed upon the externally driven accelerating fields in the cavity. The spawning of electromagnetic cavity modes as wakefields from the passing beam is analogous to a drumstick striking a drumhead and exciting many resonant mechanical modes.

The beam wakefields in an RF cavity excite a subset of the spectrum of the many electromagnetic modes, including the externally driven TM₀₁ mode. There are then a host of beam instabilities that can occur as the repetitive particle beam passes through the RF cavity, each time adding to the wakefield energy in a collection of modes.

For a particle bunch with charge q, a length much shorter than the wavelength of a given cavity mode, and traversing the cavity at time t=0, the amplitude of the wakefield voltage left behind in the cavity in a given mode is given by

,

where:

R is the shunt impedance of the cavity mode defined by

,

E is the electric field of the RF mode,

P_d is the power dissipated in the cavity to produce the electric field E,

Q_L is the "loaded Q" of the cavity, which takes into account energy leakage out of the coupling antenna,

ω_o is the angular frequency of the mode,

the imaginary exponential is the mode's sinusoidal time variation,

the real exponential term quantifies the decay of the wakefield with time, and

is termed the loss parameter of the RF mode.

The shunt impedance R can be calculated from the solution of the electromagnetic fields of a mode, typically by a computer program that solves for the fields. In the equation for V_wake, the ratio R/Q_o serves as a good comparative measure of wakefield amplitude for various cavity shapes, since the other terms are typically dictated by the application and are fixed. Mathematically,

,

where relations defined above have been used. R/Q_o is then a parameter that factors out cavity dissipation and is viewed as measure of the cavity geometry's effectiveness of producing accelerating voltage per stored energy in its volume. The wakefield being proportional to R/Q_o can be seen intuitively since a cavity with small beam apertures concentrates the electric field on axis and has high R/Q_o, but also clips off more of the particle bunch's radiation field as deleterious wakefields.

The calculation of electromagnetic field buildup in a cavity due to wakefields can be complex and depends strongly on the specific accelerator mode of operation. For the straightforward case of a storage ring with repetitive particle bunches spaced by time interval T_b and a bunch length much shorter than the wavelength of a given mode, the long term steady state wakefield voltage presented to the beam by the mode is given by

,

where:

is the decay of the wakefield between bunches, and

δ is the phase shift of the wakefield mode between bunch passages through the cavity.

As an example calculation, let the phase shift δ=0, which would be close to the case for the TM₀₁ mode by design and unfortunately likely to occur for a few HOM's. Having δ=0 (or an integer multiple of an RF mode's period, δ=n2π) gives the worse-case wakefield build-up, where successive bunches are maximally decelerated by previous bunches' wakefields and give up even more energy than with only their "self wake". Then, taking ω = 2π 500 MHz, T_b=1 µs, and Q_L=106, the buildup of wakefields would be V_{ss wake}=637×V_wake. A pitfall for any accelerator cavity would be the presence of what is termed a "trapped mode". This is an HOM that does not leak out of the cavity and consequently has a Q_L that can be orders of magnitude larger than used in this example. In this case, the buildup of wakefields of the trapped mode would likely cause a beam instability. The beam instability implications due to the V_{ss wake} wakefields is thus addressed differently for the fundamental accelerating mode TM₀₁ and all other RF modes, as described next.