19  Wakefields

19.1 Geometric Wakefields

The legacy wakefield interface supports two main routes:

1. compute the wakefield of a round metallic beam pipe from an impedance model
2. import a discretized wake function from file

For a round metallic beam pipe with radius a, OPAL distinguishes DC and AC conductivity models. The classical Drude-style conductivities are

\[ \sigma_{\mathrm{DC}} = \frac{n e^2 \tau}{m} \tag{19.1}\]

and

\[ \sigma_{\mathrm{AC}} = \frac{\sigma_{\mathrm{DC}}}{1 - i \omega \tau}. \tag{19.2}\]

With these ingredients, the longitudinal impedance can be transformed back into the longitudinal wake. For the DC case the impedance model is

\[ Z_{L,\mathrm{DC}}(k) = \frac{1}{c a} \frac{2}{\lambda/k - i k a / 2} \tag{19.3}\]

with

\[ \lambda = \sqrt{\frac{2 \pi \sigma |k|}{c}} \left(i + \operatorname{sign}(k)\right). \tag{19.4}\]

The longitudinal wake is then obtained from the inverse cosine transform of the real part of the impedance,

\[ W_L(s) = 10^{-12} \frac{2 c}{\pi} \Re \left( \int_0^\infty \Re(Z_L(k)) \cos(k s)\, dk \right), \tag{19.5}\]

and the transverse wake follows through the Panofsky-Wenzel relation.

OPAL evaluates these integrals numerically with Simpson integration on an equidistant wave-number mesh. Since this is usually done once during initialization, the computational overhead is typically acceptable.

19.2 CSR Wakefields

The CSR model in OPAL-T is built around one-dimensional steady-state or quasi-steady-state approximations. The derivation starts from the Liénard-Wiechert fields of particles moving on a circular orbit and leads to impedance and wakefield models that act on trailing particles only.

The legacy manual distinguishes:

• 1D-CSR
• 1D-CSR-IGF

The 1D-CSR model follows the staged bend-entry, steady-state, and exit picture of Saldin, Schneidmiller, and Yurkov. The 1D-CSR-IGF variant uses an integrated Green function for faster wake evaluation.

Both models neglect transverse structure and several transient corrections, but they capture the dominant longitudinal CSR effect for short bunches in many beamline studies.

19.3 The WAKE Command

WAKE defines the wakefield data attached to an element.

General syntax:

label: WAKE, TYPE=string, NBIN=integer, CONST_LENGTH=bool,
             CONDUCT=string, Z0=real, FORM=string, RADIUS=real,
             SIGMA=real, TAU=real, FNAME=string, FILTERS={...};

For CSR wakes the reduced syntax is:

label: WAKE, TYPE=string, NBIN=integer, FILTERS={...};

19.4 The FILTER Command

FILTER defines a smoothing or differentiation filter applied to the longitudinal histogram of the bunch.

Syntax:

label: FILTER, TYPE=string, NFREQ=integer, THRESHOLD=real,
               NPOINTS=integer, NLEFT=integer, NRIGHT=integer,
               POLYORDER=integer;