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:toc:
[[chp:opalcycl]]
:stem: latexmath
:sectnums:
[[chp:wakefields]]
Wakefields
----------
_OPAL-t_ provides methods to compute CSR and short-range geometric
wakefields.
[[sec:wakefields:shortrange]]
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 latexmath:[$a$] 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
latexmath:[\[\sigma_{DC} = \frac{ne^2\tau}{m} \label{eq:dc_cond}\]] with
latexmath:[$n$] the density of conduction electrons with charge
latexmath:[$e$], latexmath:[$\tau$] the relaxation time, and
latexmath:[$m$] the electron mass. The `AC` conductivity, a response to
applied oscillation fields, is given by
latexmath:[\[\sigma_{AC} = \frac{\sigma_{DC}}{1-i\omega\tau} \label{eq:ac_cond}\]]
with latexmath:[$\omega$] denoting the frequency of the fields.
The longitudinal impedance with `DC` conductivity is given by
latexmath:[\[\label{eq:Z[2]}
Z_{Ldc}(k) = \dfrac{1}{ca} \dfrac{2}{\frac{\lambda}{k}-\frac{ika}{2}}\]]
where
latexmath:[\[\lambda=\sqrt{\dfrac{2\pi\sigma \vert k\vert}{c}}(i+sign(k))\]]
with latexmath:[$c$] denoting the speed of light and latexmath:[$k$] the
wave number.
The longitudinal wake can be obtained by an inverse Fourier
transformation of the impedance. Since latexmath:[$Re(Z_L(k))$] drops at
high frequencies faster than latexmath:[$Im(Z_L(k))$] the cosine
transformation can be used to calculate the wake. The following equation
holds in both, the `DC` and `AC`, case latexmath:[\[\label{eq:Calc_Wl}
W_L(s)=10^{-12} \dfrac{2c}{\pi}Re\left(\int_0^\infty Re(Z_L(k))\cos (ks)dk\right)\]]
with latexmath:[$Z_L(k)$] either representing
latexmath:[$Z_{L_{DC}}(k)$] or latexmath:[$Z_{L_{AC}}(k)$] depending on
the conductivity. With help of the Panofsky-Wenzel theorem
latexmath:[\[Z_L(k) = \frac{k}{c}Z_T(k).\]] we can deduce the transverse
wakefield from Equation [Calc_Wl]: latexmath:[\[\label{eq:Calc_Wt}
W_T(s)= 10^{-12} \dfrac{2c}{\pi}Re\left(\int_0^\infty Re( \frac{c}{k}Z_L(k))\cos (ks)dk\right).\]]
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 latexmath:[$\Delta k$] with a
big latexmath:[$N$] 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.
[[csr-wakefields]]
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`.
[[sec:wakecmd]]
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;
....
.Wakefield command summary
[cols="<,<",options="header",]
|=======================================================================
|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 [latexmath:[$\Omega$]]
|`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
[latexmath:[$\Omega^{-1} m$]]
|`TAU` |Material constant dependent on the beam pipe material in
[latexmath:[$s$]]
|`FNAME` |Specify a file that provides a wake function
|`FILTER` |The names of the filters that should be applied
|=======================================================================
[[sec:wakefield]]
Define the Wakefield to be used
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The `WAKE` command defines data for a wake function on an element
see Section [Element:common].
[[sec:WTYPE]]
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`.
[[sec:NBIN]]
Define the number of bins
~~~~~~~~~~~~~~~~~~~~~~~~~
The number of bins used in the calculation of the line density.
[[sec:CONSTLEN]]
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.
[[sec:CONDUCT]]
Define the conductivity
~~~~~~~~~~~~~~~~~~~~~~~
The conductivity of the bunch which can be set to either `AC` or `DC`.
This has no effect on CSR wakefunctions.
[[sec:Z]]
Define the impedance
~~~~~~~~~~~~~~~~~~~~
The impedance latexmath:[$Z_0$] of the beam pipe in
[latexmath:[$\Omega$]]. This has no effect on CSR wakefunctions.
[[sec:FORM]]
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.
[[sec:RADIUS]]
Define the radius of the beam pipe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The radius of the beam pipe in [m]. This has no effect on CSR
wakefunctions.
[[sec:SIGMA]]
Define the sigma of the beam pipe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The latexmath:[$\sigma$] of the beam pipe (material constant), see
Equation [dc_cond]. This has no effect on CSR wakefunctions.
[[sec:TAU]]
Define the relaxation time (tau) of the beam pipe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The latexmath:[$\tau$] 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.
[[sec:WFNAME]]
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.
[[sec:FILTER]]
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.
[[the-filter-command]]
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:
latexmath:[\[f_i = \frac{7\cdot f_{i-4} + 24\cdot f_{i-2} + 34\cdot f_{i} + 24\cdot f_{i+2} + 7\cdot f_{i+4}}{96}\]]
and
latexmath:[\[f_i = \frac{7\cdot f_{i-2} + 24\cdot f_{i-1} + 34\cdot f_{i} + 24\cdot f_{i+1} + 7\cdot f_{i+2}}{96}.\]]
For the derivative a standard second order stencil is used:
latexmath:[\[f'_i = \frac{f_{i-2} - 8\cdot f_{i-1} + 8\cdot f_{i+1} - f_{i+2}}{h}\]]
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): latexmath:[\[g_{i} =
\begin{cases}
i \frac{2\pi \imath}{N\cdot L} & i < N/2 \\
-i \frac{2\pi \imath}{N\cdot L} & i > N/2
\end{cases}\]] where latexmath:[$N$] is the total number of
coefficients/sampling points and latexmath:[$L$] is the length of the
bunch.