Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with
or
.
Download ZIP
Browse files

insert commentary from JAP paper to relevant places

  • Loading branch information...
commit efe5d2accef3ce1e2af73e568b86ff45fdf08871 1 parent be34493
@jberger authored
View
1  bibliography.bib
@@ -475,7 +475,6 @@ @article{mcdonald_design_1988
@article{zawadzka_evanescent_2001,
author = {Justyna Zawadzka and Dino A. Jaroszynski and John J. Carey and Klaas Wynne},
-collaboration = {},
title = {Evanescent-wave acceleration of ultrashort electron pulses},
publisher = {AIP},
year = {2001},
View
13 inc/external_forces/external_forces.tex
@@ -2,6 +2,17 @@
\subsection{Extension to Include External Forces} \label{sec:external_forces}
+\subsubsection{Validity and Limitations}
+
+The presented extension to the AG model to include external forces acting on the electron pulse is valid only within the limits of the analytical method itself; in particular, its mean internal space-charge field and self-similar Gaussian approximations.\cite{michalik_analytic_2006}
+As a result, the extension reflects a first-order (i.e., linear force) analysis of the effects of electron optics upon electron pulse propagation.
+Nonetheless, for free-space propagation, the AG model of charge bunch dynamics has already been shown to be very consistent with full Monte Carlo (i.e., particle tracking) simulations for a wide variety of electron pulse shapes,\cite{michalik_analytic_2006,michalik_evolution_2009} including the uniform ellipsoid.\cite{luiten_how_2004}
+This successful benchmarking is due primarily to the versatility of the AG model which results from its use of transverse and longitudinal pulse position and momentum variances.
+Consequently, the AG approach is applicable to both the single electron per pulse limit,\cite{lobastov_four-dimensional_2005} where momentum variances determine the pulse evolution and the model is exact (obeying Gaussian optics), and the high charge density limit in which space-charge effects dominate.\cite{luiten_how_2004,siwick_ultrafast_2002,cao_femtosecond_2003}
+It is this versatility combined with its computational efficiency that makes the presented extended AG model particularly suitable for rapid initial assessments of pulsed electron microscope column designs and electron pulse delivery systems in UED experiments.
+Verification of the validity (and determination of the limits) of the extended AG model will, of course, require future comparison with both experiment and more complete simulations of electron pulse propagation dynamics (e.g., full particle tracking models) that include nonlinear forces, for both the intra-pulse space-charge field and the description of aberrations in electron optics.
+
+\subsubsection{Mathematical Formulation of External Force Contribution}
\ref{eq:potential_integral} is the entry point for the internal Coulombic repulsion force; since both potentials and integrals are linear, one may add additional potentials so that it becomes
\begin{equation}
\begin{split}
@@ -26,7 +37,7 @@ \subsection{Extension to Include External Forces} \label{sec:external_forces}
\frac{\partial}{\partial t} A^{-1}_{ij} = K^{flow}_{ij} + K^{int}_{ij} + \sum K^{ext}_{ij}
\end{equation}
-\subsection{Specific Forms of External Forces}
+\subsubsection{Specific Forms of External Forces}
If $\vec{F}$ is a static field given by $\frac{qV}{d}\hat{z}$ then $K^{ext} = \hat{0}$. However, if the external force is a lensing field given by $\vec{F} = -M\cdot(x\hat{x}+y\hat{y})$ then
\begin{equation}
K^{ext} = M \cdot
View
3  inc/initial_conditions/initial_conditions.tex
@@ -17,7 +17,8 @@ \subsection{Determining Appropriate Initial Conditions} \label{sec:initial_condi
\sigma_{ z } ( 0 ) = ( \Delta z)^{2} = \frac{ ( v_{{ \scriptscriptstyle 0}} \tau )^{2} }{ 2 } \text{ ,}
\end{equation}
with the velocity of the pulse in the lab frame $v_{{ \scriptscriptstyle 0}} = \sqrt{ 2 e V / m_{e} d } $, where $ V $ and $ d $ are the potential and cathode-anode separation of the gun's DC acceleration region respectively.
-For spatially-uniform photoemission from a planar photocathode, we have $ \gamma_{\smallT} ( 0 ) = 0 $; i.e., there is no initial spatial momentum chirp in the electron bunch. Immediately after photo-generation, however, it is strictly inaccurate to say that $ \gamma_{ z } ( 0 ) = 0 $.
+For spatially-uniform photoemission from a planar photocathode, we have $ \gamma_{\smallT} ( 0 ) = 0 $; i.e., there is no initial spatial momentum chirp in the electron bunch.
+Immediately after photo-generation, however, it is strictly inaccurate to say that $ \gamma_{ z } ( 0 ) = 0 $.
This is because the excess photoemission energy $ \Delta E = \hbar \omega - \Phi $, associated with the electron emission from a material with work function $\Phi$ using photons of energy $ \hbar \omega $ (which gives the maximum initial velocity of an electron, $ v_{max} = \sqrt{ 2 \Delta E/m_{e} } $), can produce an initial momentum chirp.
Nonetheless, the initial non-zero nature of $ \gamma_{z} $ is quickly dominated by the evolution of the pulse, so this approximation will suffice in the region far from the gun.
View
2  inc/liouvilles_theorem/liouvilles_theorem.tex
@@ -29,3 +29,5 @@ \subsection{Implications of Liouville's Theorem}
This will become more apparent when in future sections
%TODO include internal reference
it is shown that at the focal point of a lens, $\gamma_{\alpha}$ goes through zero.
+
+In the transverse dimension, this is consistent with the definition of `coherent fluence' employed by Reed et al.\cite{reed_evolution_2009} for time-resolved electron microscopy and, of course, the spatial emittance of an electron beam\cite{jensen_theoretical_2006,siwick_ultrafast_2002}.
View
6 inc/ms_model/ms_model.tex
@@ -235,3 +235,9 @@ \subsection{The Differential Equation System}
\frac{d\eta_{\alpha}}{dt} = - \frac{2 \gamma_{\alpha} \eta_{\alpha}}{m \sigma_{\alpha}}
\end{gather}
\end{subequations}
+
+\subsection{Validity and Limitations}
+
+This AG formalism of Michalik and Sipe\cite{michalik_analytic_2006} employs a mean-field approximation in the evolution of the dynamical effect of the electron pulse's internal space-charge field.
+This is of course not strictly accurate for a Gaussian charge distribution and the error leads to a distortion of the pulse shape.
+Even so, the self-similar AG model of charge bunch dynamics has been benchmarked against particle tracking simulations for a number of electron pulse shapes,\cite{michalik_analytic_2006,michalik_evolution_2009} including the uniform ellipsoid which explicitly features a linear internal space-charge field.\cite{luiten_how_2004}
Please sign in to comment.
Something went wrong with that request. Please try again.