... | ... | @@ -340,7 +340,10 @@ As the name implies, the `GAUSS` distribution type can generate |
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distributions with a general Gaussian shape (here we show a
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one-dimensional example):
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latexmath:[\[f(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{(x - \bar{x})^{2}}{2 \sigma_{x}^{2}}}\]]
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[latexmath]
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++++
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f(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{(x - \bar{x})^{2}}{2 \sigma_{x}^{2}}}
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++++
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where latexmath:[\bar{x}] is the average value of latexmath:[x].
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However, the `GAUSS` distribution can also be used to generate an
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... | ... | @@ -441,18 +444,21 @@ Name:DISTRIBUTION, TYPE = GAUSS, |
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....
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This creates a Gaussian shaped distribution with zero transverse
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emittance, zero energy spread, latexmath:[\sigma_{x} = {1.0}{mm}],
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latexmath:[\sigma_{y} = {3.0}{mm}],
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latexmath:[\sigma_{z} = {2.0}{mm}] and an average energy of:
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emittance, zero energy spread, latexmath:[\sigma_{x} = {1.0}\mathrm{mm}],
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latexmath:[\sigma_{y} = {3.0}\mathrm{mm}],
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latexmath:[\sigma_{z} = {2.0}\mathrm{mm}] and an average energy of:
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latexmath:[\[W = {1.2}{MeV}\]]
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[latexmath]
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++++
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W = {1.2}{MeV}
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++++
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In the latexmath:[x] direction, the Gaussian distribution is cutoff at
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latexmath:[x = 2.0 \times \sigma_{x} = {2.0}{mm}]. In the
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latexmath:[x = 2.0 \times \sigma_{x} = {2.0}\mathrm{mm}]. In the
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latexmath:[y] direction it is cutoff at
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latexmath:[y = 2.0 \times \sigma_{y} = {6.0}{mm}]. This distribution
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latexmath:[y = 2.0 \times \sigma_{y} = {6.0}\mathrm{mm}]. This distribution
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is _injected_ into the simulation at an average position of
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latexmath:[(\bar{x},\bar{y},\bar{z})=({1.0}{mm}, {-2.0}{mm}, {10.0}{mm})].
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latexmath:[(\bar{x},\bar{y},\bar{z})=({1.0}\mathrm{mm}, {-2.0}\mathrm{mm}, {10.0}\mathrm{mm})].
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[[sec:gaussdisttypephotoinjector]]
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`GAUSS` Distribution for Photoinjector
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... | ... | @@ -479,8 +485,8 @@ imposed on the flat top, latexmath:[t_\mathrm{flattop}], in |
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Figure [flattop].
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|=======================================================================
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image:./figures/flattop.png[_OPAL_ emitted `GAUSS` distribution with
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flat top.,scaledwidth=80.0%]
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.Figure 1: _OPAL_ emitted `GAUSS` distribution with flat top.
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image:./figures/flattop.png[width=700]
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A useful feature of the `GAUSS` distribution type is the ability to
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mimic the initial distribution from a photoinjector. For this purpose we
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... | ... | @@ -492,26 +498,35 @@ measured laser profiles, `TRISE` and `TFALL` from |
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Table [distattremittedgauss] do not define RMS quantities, but instead
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are given by (See also Figure [flattop]):
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latexmath:[\[\begin{aligned}
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[latexmath]
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++++
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\begin{aligned}
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\texttt{TRISE} = t_{R} &= \left(\sqrt{2 \ln(10)} - \sqrt{2 \ln \left(\frac{10}{9} \right)} \right) \sigma_{R}\\
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& = 1.6869 \sigma_{R} \\
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\texttt{TFALL} = t_{F} &= \left(\sqrt{2 \ln(10)} - \sqrt{2 \ln \left(\frac{10}{9} \right)} \right) \sigma_{F}\\
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& = 1.6869 \sigma_{F}\end{aligned}\]]
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& = 1.6869 \sigma_{F}\end{aligned}
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++++
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where latexmath:[\sigma_{R}] and latexmath:[\sigma_{F}] are the
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Gaussian, RMS rise and fall times respectively. The flat-top portion of
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the profile, `TPULSEFWHM`, is defined as (See also Figure [flattop]):
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latexmath:[\[\texttt{TPULSEFWHM} = \mathrm{FWHM}_{P} = t_\mathrm{flattop} + \sqrt{2 \ln 2} \left( \sigma_{R} + \sigma_{F} \right)\]]
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[latexmath]
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++++
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\texttt{TPULSEFWHM} = \mathrm{FWHM}_{P} = t_\mathrm{flattop} + \sqrt{2 \ln 2} \left( \sigma_{R} + \sigma_{F} \right)
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++++
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Total emission time, latexmath:[t_{E}], of this distribution, is a
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function of the longitudinal cutoff, `CUTOFFLONG`
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see Table [distattrgauss], and is given by:
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latexmath:[\[\begin{aligned}
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[latexmath]
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++++
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\begin{aligned}
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t_{E}(\texttt{CUTOFFLONG}) &= \mathrm{FWHM}_{P} - \frac{1}{2} (\mathrm{FWHM}_{R} + \mathrm{FWHM}_{F})
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+ \texttt{CUTOFFLONG} (\sigma_{R} + \sigma_{F}) \\
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&= \mathrm{FWHM}_{P} + \frac{\texttt{CUTOFFLONG} - \sqrt{2 \ln 2}}{1.6869} (\texttt{TRISE} + \texttt{TFALL})\end{aligned}\]]
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&= \mathrm{FWHM}_{P} + \frac{\texttt{CUTOFFLONG} - \sqrt{2 \ln 2}}{1.6869} (\texttt{TRISE} + \texttt{TFALL})\end{aligned}
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++++
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Finally, we can also impose oscillations over the flat-top portion of
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the laser pulse in Figure [flattop], latexmath:[t_\mathrm{flattop}].
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... | ... | @@ -571,18 +586,21 @@ correlate latexmath:[R]. The correlation coefficient matrix |
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latexmath:[\sigma] in latexmath:[x], latexmath:[p_x],
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latexmath:[z], latexmath:[p_z] phase space reads:
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latexmath:[\[\sigma= \left[
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[latexmath]
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++++
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\sigma= \left[
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\begin{array}{cccc}
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1 &c_x&r51 &r61\\
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c_x&1 &r52 &r62\\
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r51 &r52 &1 &c_t\\
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r61 &r62 &c_t &1\\
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\end{array}
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\right] \\\]]
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\right] \\
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++++
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The Cholesky decomposition of the symmetric positive-definite matrix
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latexmath:[\sigma] is latexmath:[\sigma=C^{\mathbf{T}}C], then the
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correlated distribution is latexmath:[\transpose{C}R].
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correlated distribution is latexmath:[C^{\mathbf{T}}R].
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*Note*: Correlations work for the moment only with the Gaussian
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distribution and are experimental, so there are no guarantees as to its
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... | ... | @@ -594,14 +612,17 @@ take care. |
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As an example of defining a correlated beam, let the initial correlation
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coefficient matrix be:
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latexmath:[\[\sigma= \left[
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[latexmath]
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++++
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\sigma= \left[
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\begin{array}{cccc}
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1 &0.756 &0.023 &0.496\\
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0.756 &1 &0.385 &-0.042\\
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0.023 &0.385 &1 &-0.834\\
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0.496 &-0.042 &-0.834 &1\\
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\end{array}
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\right]\]]
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\right]
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++++
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then the corresponding distribution command will read:
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... | ... | @@ -872,17 +893,23 @@ option as listed in Table [distattremitmodelnoneastra]. However, in this |
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case, the energy defined by the `EKIN` attribute is added to each
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emitted particle’s momentum in a random way:
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latexmath:[\[\begin{aligned}
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[latexmath]
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++++
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\begin{aligned}
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p_{total} &= \sqrt{\left(\frac{\texttt{EKIN}}{mc^{2}} + 1\right)^{2} - 1} \\
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p_{x} &= p_{total} \sin(\phi) \cos(\theta)) \\
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p_{y} &= p_{total} \sin(\phi) \sin(\theta)) \\
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p_{z} &= p_{total} |{\cos(\theta)}|
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\end{aligned}\]]
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\end{aligned}
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++++
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where latexmath:[\theta] is a random angle between latexmath:[0] and
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latexmath:[\pi], and latexmath:[\phi] is given by
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latexmath:[\[\phi = 2.0 \arccos \left( \sqrt{x} \right)\]]
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[latexmath]
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++++
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\phi = 2.0 \arccos \left( \sqrt{x} \right)
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++++
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with latexmath:[x] a random number between latexmath:[0] and
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latexmath:[1].
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