wakefields

Table of Contents

1. Wakefields

1. Wakefields

OPAL-t provides methods to compute CSR and short-range geometric wakefields.

1.1. Geometric Wakefields

Basically there are two different kind of wakefields that can be used. The first one is the wakefield of a round, metallic beam pipe that can be calculated numerically (see Sections [sec:wakefield] - [sec:TAU]). Since this also limits the applications of wakefields we also provide a way to import a discretized wakefield from a file The wakefield of a round, metallic beam pipe with radius a can be calculated by inverse FFT of the beam pipe impedance. There are known models for beam pipes with DC and AC conductivity. The DC conductivity of a metal is given by

\sigma_{DC} = \frac{ne^2\tau}{m}

with n the density of conduction electrons with charge e, \tau the relaxation time, and m the electron mass. The AC conductivity, a response to applied oscillation fields, is given by

\sigma_{AC} = \frac{\sigma_{DC}}{1-i\omega\tau}

with \omega denoting the frequency of the fields.

The longitudinal impedance with DC conductivity is given by

 Z_{Ldc}(k) = \dfrac{1}{ca} \dfrac{2}{\frac{\lambda}{k}-\frac{ika}{2}}

where

\lambda=\sqrt{\dfrac{2\pi\sigma \vert k\vert}{c}}(i+sign(k))

with c denoting the speed of light and k the wave number.

The longitudinal wake can be obtained by an inverse Fourier transformation of the impedance. Since Re(Z_L(k)) drops at high frequencies faster than Im(Z_L(k)) the cosine transformation can be used to calculate the wake. The following equation holds in both, the DC and AC, case

W_L(s)=10^{-12} \dfrac{2c}{\pi}Re\left(\int_0^\infty Re(Z_L(k))\cos (ks)dk\right)

with Z_L(k) either representing Z_{L_{DC}}(k) or Z_{L_{AC}}(k) depending on the conductivity. With help of the Panofsky-Wenzel theorem

Z_L(k) = \frac{k}{c}Z_T(k).

we can deduce the transverse wakefield from Equation [Calc_Wl]:

W_T(s)= 10^{-12} \dfrac{2c}{\pi}Re\left(\int_0^\infty Re( \frac{c}{k}Z_L(k))\cos (ks)dk\right).

To calculate the integrals in Equation [Calc_Wl,Calc_Wt] numerically the Simpson integration schema with equidistant mesh spacing is applied. This leads to an integration with small \Delta k with a big N which is computational not optimal with respect to efficiency. Since we calculate the wakefield usually just once in the initialization phase the overall performance will not be affected from this.

1.2. CSR Wakefields

The electromagnetic field due to a particle moving on a circle in free space can be calculated exactly with the Liénard-Wiechert potentials. The field has been calculated for all points on the same circle [bib:schott,bib:schwinger1949]. For high particle energies the radiated power is almost exclusively emitted in forward direction, whereas for low energies a fraction is also emitted in transverse and backward direction. For the case of high-energetic particles an impedance in forward direction can be calculated [bib:murphy1997]. The procedure is then the same as for a regular wakefield with the important difference that wakes exert forces on trailing particles only. The electromagnetic fields of a particle propagating on the mid-plane between two parallel metallic plates that stretch to infinity [bib:schwinger1949] and for finite plates [bib:nodvick1954] can also be calculated. For the infinite plates an impedance can be calculated [bib:murphy1997].

All of these approaches for CSR neglect any transient effects due to the finite length of the bend. Instead they describe the steady state case of a bunch circling infinitely long in the field of a dipole magnet. In [bib:saldin1997] the four different stages of a bunch passing a bending magnet are treated separately and for each a corresponding wake function is derived. This model is used in OPAL-t for 1D-CSR.

The 1-dimensional approach also neglects any influence of the transverse dimensions and of changes in current density between retarded and current time. On the other hand it gives a good approximation of effects due to CSR in short time.

In addition to the 1D-CSR model also one that makes use of an integrated Green function [bib:mitchell2013], 1D-CSR-IGF.

1.3. The `WAKE` Command

The general input format is

label:WAKE, TYPE=string, NBIN=real, CONST_LENGTH=bool,
      CONDUCT=string, Z0=real, FORM=string, RADIUS=real,
      SIGMA=real, TAU=real, FILTERS=string-array;

The format for a CSR wake is

label:WAKE, TYPE=string, NBIN=real, FILTERS=string-array;

Table 1. Wakefield command summary
Command	Purpose
`WAKE`	Specify a wakefield
`TYPE`	Specify the wake function [`1D-CSR`, `1D-CSR-IGF`, `LONG-SHORT-RANGE`, `TRANSV-SHORT-RANGE`, `LONG-TRANSV-SHORT-RANGE`]
`NBIN`	Number of bins used in the calculation of the line density
`CONST_LENGTH`	`TRUE` if the length of the bunch is considered to be constant
`CONDUCT`	Conductivity [`AC`, `DC`]
`Z0`	Impedance of the beam pipe in [`\Omega`]
`FORM`	The form of the beam pipe [`ROUND`]
`RADIUS`	The radius of the beam pipe in [m]
`SIGMA`	Material constant dependent on the beam pipe material in [`\Omega^{-1} m`]
`TAU`	Material constant dependent on the beam pipe material in [`s`]
`FNAME`	Specify a file that provides a wake function
`FILTER`	The names of the filters that should be applied

1.4. Define the Wakefield to be used

The WAKE command defines data for a wake function on an element see Section [Element:common].

1.5. Define the wakefield type

A string value see Section [astring] to specify the wake function, either 1D-CSR, 1D-CSR-IGF, LONG-SHORT-RANGE, TRANSV-SHORT-RANGE or LONG-TRANSV-SHORT-RANGE.

1.6. Define the number of bins

The number of bins used in the calculation of the line density.

1.7. Define the bunch length to be constant

With the CONST_LENGTH flag the bunch length can be set to be constant. This has no effect on CSR wakefunctions.

1.8. Define the conductivity

The conductivity of the bunch which can be set to either AC or DC. This has no effect on CSR wakefunctions.

1.9. Define the impedance

The impedance Z_0 of the beam pipe in [\Omega]. This has no effect on CSR wakefunctions.

1.10. Define the form of the beam pipe

The form of the beam pipe can be set to ROUND. This has no effect on CSR wakefunctions.

1.11. Define the radius of the beam pipe

The radius of the beam pipe in [m]. This has no effect on CSR wakefunctions.

1.12. Define the sigma of the beam pipe

The \sigma of the beam pipe (material constant), see Equation [dc_cond]. This has no effect on CSR wakefunctions.

1.13. Define the relaxation time (tau) of the beam pipe

The \tau defines the relaxation time and is needed to calculate the impedance of the beam pipe see Equation [dc_cond]. This has no effect on CSR wakefunctions.

1.14. Import a wakefield from a file

Since we only need values of the wake function at several discreet points to calculate the force on the particle it is also possible to specify these in a file.To get required data points of the wakefield not provide in the file we linearly interpolate the available function values. The files are specified in the SDDS format [bib:borland1995,bib:borland1998].

Whenever a file is specified OPAL will use the wakefield found in the file and ignore all other commands related to round beam pipes.

1.15. List of Filters

Array of names of filters to be applied to the longitudinal histogram of the bunch to get rid of the noise and to calculate derivatives. All the filters are applied to the line density in the order they appear in the array. The last filter is also used for calculating the derivatives. The actual filters have to be defined elsewhere.

1.16. The `FILTER` Command

Filters can be defined which then are applied to the line density of the bunch. The following smoothing filters are implemented: Savitzky-Golay, Stencil, FixedFFTLowPass, RelativFFTLowPass. The input format for them is

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

TYPE: The type of filter: Savitzky-Golay, Stencil, FixedFFTLowPass, RelativFFTLowPass
NFREQ: Only used in FixedFFTLowPass: the number of frequencies to keep
THRESHOLD: Only used in RelativeFFTLowPass: the minimal strength of frequency compared to the strongest to consider.
NPOINTS: Only used in Savitzky-Golay: width of moving window in number of points
NLEFT: Only used in Savitzky-Golay: number of points to the left
NRIGHT: Only used in Savitzky-Golay: number of points to the right
POLYORDER: Only used in Savitzky-Golay: polynomial order to be used in least square approximation

The Savitzky-Golay filter and the ones based on the FFT routine provide a derivative on a natural way. For the Stencil filter a second order stencil is used to calculate the derivative.

An implementation of the Savitzky-Golay filter can be found in the Numerical Recipes. The Stencil filter uses the following two stencil consecutively to smooth the line density:

f_i = \frac{7\cdot f_{i-4} + 24\cdot f_{i-2} + 34\cdot f_{i} + 24\cdot f_{i+2} + 7\cdot f_{i+4}}{96}

and

f_i = \frac{7\cdot f_{i-2} + 24\cdot f_{i-1} + 34\cdot f_{i} + 24\cdot f_{i+1} + 7\cdot f_{i+2}}{96}.

For the derivative a standard second order stencil is used:

f'_i = \frac{f_{i-2} - 8\cdot f_{i-1} + 8\cdot f_{i+1} - f_{i+2}}{h}

This filter was designed by Ilya Pogorelov for the ImpactT implementation of the CSR 1D model.

The FFT based smoothers calculate the Fourier coefficients of the line density. Then they set all coefficients corresponding to frequencies above a certain threshold to zero. Finally the back-transformation is calculate using this coefficients. The two filters differ in the way they identify coefficients which should be set to zero. FixedFFTLowPass uses the n lowest frequencies whereas RelativeFFTLowPass searches for the coefficient which has the biggest absolute value. All coefficients which, compared to this value, are below a threshold (measure in percents) are set to zero. For the derivative the coefficients are multiplied with the following function (this is equivalent to a convolution):

g_{i} =
\begin{cases}
i \frac{2\pi \imath}{N\cdot L} & i < N/2 \\
-i \frac{2\pi \imath}{N\cdot L} & i > N/2
\end{cases}

where N is the total number of coefficients/sampling points and L is the length of the bunch.