Changeset 579 in ETALON

Timestamp:

Apr 28, 2016, 4:39:20 PM (8 years ago)

Author:

malovyts

Message:

Removed comments

Location:

papers/2016_IPAC/2016_IPAC_Malovytsia_ModelComparison

Files:

• MOPMB004.pdf (modified) (previous)
• MOPMB004.tex (modified) (8 diffs)

papers/2016_IPAC/2016_IPAC_Malovytsia_ModelComparison/MOPMB004.tex

@@ -85 +85 @@
                 and shown that they are in agreement 
                 within the experimental errors. 
-                %To have a better agreement 
-                %between predictions and experimental measurements 
-                %we also report on the interference effects 
-                %that modulate the signal in the near-field zone.
         \end{abstract}
-        
-        
-        
-        \section{Introduction}
-        
+
+%=======================================================================================================        
+\section{Introduction}
         The production and measurement of sub-picosecond bunches is an important topic for modern accelerators. 
-%       Recent advance in accelerating technologies introduced possibility of creating subpicosecond bunches,
-%        which are required for the free-electron lasers~\cite{FELEvt} and the plasma accelerators~\cite{PLASMABerry06}.
-%        Such short bunches confront us with a challenge of measuring their length reliably in a non-destructive way.
-%        Leaving aside methods, that destroy important parameters of the bunch,
-To measure reliably the length of such short bunches with destroying them several approaches are possible:
-         \begin{itemize}
-          \item Electro-Optic (EO) sampling~\cite{Fitch99} uses a non linear crystal in which the bunch wakefield will induce optical changes. It requires a femtosecond laser. Its limitations due to material properties are discussed in~\cite{EOLimSteff09}.
-          \item Coherent Transition Radiation (CTR)~\cite{Lai94} uses the radiation emitted when the beam crosses a thin foil. In some cases it may be difficult to discriminate the signal from CTR for other sources of radiation (e.g.: synchrotron radiation) generated further upstream.
-         \item Coherent Smith-Purcell Radiation~\cite{Nguyen97} (CSPR),
-           uses a grating to induce the emission of radiation. It 
-           has the advantage of dispersing the radiation at the point of emission and therefore being more immune to background noise. It is described below.
+        To measure reliably the length of such short bunches with destroying them several approaches are possible:
+        \begin{itemize}
+          \item
+                  Electro-Optic (EO) sampling~\cite{Fitch99} uses a non linear crystal in which the bunch wakefield will induce optical changes. It requires a femtosecond laser. Its limitations due to material properties are discussed in~\cite{EOLimSteff09}.
+          \item
+                  Coherent Transition Radiation (CTR)~\cite{Lai94} uses the radiation emitted when the beam crosses a thin foil. In some cases it may be difficult to discriminate the signal from CTR for other sources of radiation (e.g.: synchrotron radiation) generated further upstream.
+         \item
+                  Coherent Smith-Purcell Radiation~\cite{Nguyen97} (CSPR),
+                  uses a grating to induce the emission of radiation. It 
+                  has the advantage of dispersing the radiation at the point of emission and therefore being more immune to background noise. It is described below.
         \end{itemize}
         
@@ -113 +106 @@
 
 %======================================================================================================
-        
-        \section{Principle of Smith-Purcell Radiation}
-        % geometry figure with the reference
+\section{Principle of Smith-Purcell Radiation}
         Smith-Purcell radiation is produced by a charged particle passing
-         near a surface of a conducting periodical grating.
-         In multiple papers~\cite{p020, p043, p026, p019, p039} authors considered a profile
-         of the grating as a set of the periodically repeating ``$N$'' pairs of ``rising'' an ``falling''
-         facets as shown on Fig.~\ref{fig:geom1}, with the period of repetition ``$d$'', a blaze angle
-         ``$\theta_0$'' ($\alpha_1$ in~\cite{p020, p043}), the width ``$M$'' and the length ``$L$''.
-         The choice of such profile is explained in~\cite{p020}, by a possibility to do derive
-         simpler analytical expressions and thus define the relation between the grating parameters and the SPR
-         characteristics. It is convenient to chose the same profile for a comparison purposes.
+        near a surface of a conducting periodical grating.
+        In multiple papers~\cite{p020, p043, p026, p019, p039} authors considered a profile
+        of the grating as a set of the periodically repeating ``$N$'' pairs of ``rising'' an ``falling''
+        facets as shown on Fig.~\ref{fig:geom1}, with the period of repetition ``$d$'', a blaze angle
+        ``$\theta_0$'' ($\alpha_1$ in~\cite{p020, p043}), the width ``$M$'' and the length ``$L$''.
+        The choice of such profile is explained in~\cite{p020}, by a possibility to do derive
+        simpler analytical expressions and thus define the relation between the grating parameters and the SPR
+        characteristics. It is convenient to chose the same profile for a comparison purposes.
 
         \begin{figure}
@@ -132 +123 @@
         \end{figure}
         
-        %One of the characteristic of SPR, which allows using in the profile measurement technique, is a wavelength dispersion, 
-        %       \begin{equation}\label{eq:lDispersion}
-        %               \lambda=\frac{d}{n}\left( \frac{1}{\beta} - \cos{\theta} \right)
-        %       \end{equation}
-        %where $ \lambda $ - wavelength of the radiation, in the direction of observation angle $ \theta $ (Fig.~\ref{fig:geom1}),  $ n $ - order of radiation
-        %and speed of the electron $ \beta = \sqrt{1-\frac{1}{\gamma^2}} $ expressed through the speed of light. 
-        
-        
-                
 %=======================================================================================================
-
-
-%       \begin{equation}
-%       S_{coh}=\left| \int_{0}^{\infty}Xe^{-\left( x-x_        0 \right)/\lambda_e}dx \right|^2
-%       \left| \int_{-\infty}^{\infty}Ye^{-ik_y y}dy \right|^2
-%       \left| \int_{-\infty}^{\infty}Te^{-i\omega t}dt \right|
-%       \end{equation}
-%       \begin{equation}
-%       S_{inc}=\int_{0}^{\infty}Xe^{-2(x-x_0)/\lambda_e}dx
-%       \end{equation}
-
-        \section{Single Electron Yield Models}
+\section{Single Electron Yield Models}
         The leading models to calculate the SPR Single Electron Yield (SEY) are:
         \begin{itemize}
                 
-        \item The Surface Current 
-         model~\cite{gfw}, that explains SPR through the currents that are being induced
-         on the surface of the grating by a charge passing nearby. This theory has proven to be
-         in a good agreement with experiments for energies from a few MeV to 28.5 GeV~\cite{p010, p019, p043, p026}.
+        \item
+                The Surface Current model~\cite{gfw}, that explains SPR through the currents that are being induced
+                on the surface of the grating by a charge passing nearby. This theory has proven to be
+                in a good agreement with experiments for energies from a few MeV to 28.5 GeV~\cite{p010, p019, p043, p026}.
                 \begin{equation}
-        \left( \frac{dI}{d\Omega} \right)_1=2\pi q^2 \frac{L}{d^2}\frac{1}{\lambda^3}R^2\exp{\left[-\frac{2h}{\lambda_e}\right]}
-        \end{equation}
-        \begin{equation}
-        \lambda_e=\frac{\lambda}{2\pi}\frac{\beta \gamma}{\sqrt{1+\beta^2\gamma^2\sin^2{\theta}\sin^2{\phi}}}
-        \end{equation}
-
-
-        Here, $q$ stands for the particle charge, $\lambda$ is the wavelength of the radiation emitted, $\lambda_e$ is an ``evanescent'' wavelength,
-         $\beta, \gamma$ are the velocity of the particle and its Lorentz factor,
-        $\theta, \phi$ are angles as shown on Fig.~\ref{fig:geom1}.
-         $R^2$ is a grating efficiency parameter, that depends on the radiation angle and blaze angle.
+                        \left( \frac{dI}{d\Omega} \right)_1=2\pi q^2 \frac{L}{d^2}\frac{1}{\lambda^3}R^2\exp{\left[-\frac{2h}{\lambda_e}\right]}
+                \end{equation}
+                \begin{equation}
+                        \lambda_e=\frac{\lambda}{2\pi}\frac{\beta \gamma}{\sqrt{1+\beta^2\gamma^2\sin^2{\theta}\sin^2{\phi}}}
+                \end{equation}
+
+                Here, $q$ stands for the particle charge, $\lambda$ is the wavelength of the radiation emitted, $\lambda_e$ is an ``evanescent'' wavelength,
+                $\beta, \gamma$ are the velocity of the particle and its Lorentz factor,
+                $\theta, \phi$ are angles as shown on Fig.~\ref{fig:geom1}.
+                $R^2$ is a grating efficiency parameter, that depends on the radiation angle and blaze angle.
          
-         Further in the paper, the results obtained with the expression for $R^2$ taken from~\cite{p021} will be called SC, and from the~\cite{gfw}, where the grating efficiency is calculated numerically, will be referred to as GFW.
-        
-        \item The Resonant Diffraction Radiation (RDR) model,
-         uses equation for the diffraction radiation (DR) of an electron passing near a conductive semi-plane
-          and extends it onto the case of the ``N'' periodically placed strips~\cite{p021}:
+                Further in the paper, the results obtained with the expression for $R^2$ taken from~\cite{p021} will be called SC, and from the~\cite{gfw}, where the grating efficiency is calculated numerically, will be referred to as GFW.
+        
+        \item
+                The Resonant Diffraction Radiation (RDR) model,
+                uses equation for the diffraction radiation (DR) of an electron passing near a conductive semi-plane
+                and extends it onto the case of the ``N'' periodically placed strips~\cite{p021}:
           
-          \begin{equation}\label{eq:RDR_model}
-                  \frac{d^2W_{RDR}}{d\omega d\Omega}=\frac{d^2W_{RDR}}{d\omega d\Omega}F_{n,cell}F_{N}
-          \end{equation}
-        
-        Where $\frac{d^2W_{RDR}}{d\omega d\Omega}$ is a frequency distribution of the intensity of the RDR,
-         $\frac{d^2W_{DR}}{d\omega d\Omega}$ is the frequency distribution of the intensity of the DR,
-         $F_{N}$ is a factor corresponding to the interference from $N$ strips, 
-         $F_{n,cell}$ is a term, that takes into account the interference of the DR on one strip.
+                \begin{equation}\label{eq:RDR_model}
+                        \frac{d^2W_{RDR}}{d\omega d\Omega}=\frac{d^2W_{RDR}}{d\omega d\Omega}F_{n,cell}F_{N}
+                \end{equation}
+        
+                Where $\frac{d^2W_{RDR}}{d\omega d\Omega}$ is a frequency distribution of the intensity of the RDR,
+                $\frac{d^2W_{DR}}{d\omega d\Omega}$ is the frequency distribution of the intensity of the DR,
+                $F_{N}$ is a factor corresponding to the interference from $N$ strips, 
+                $F_{n,cell}$ is a term, that takes into account the interference of the DR on one strip.
          
-         For a large number of periods one can integrate Eq.~\ref{eq:RDR_model} over the frequencies and obtain an analytical expression~(see paper~\cite{p021}) for the intensity of the SPR.
-        
-        \item The model so-called Resonant Reflection Radiation (RRR) model based on the fact that a field of a moving charged particle could be described
-         as a sum of the virtual plain waves~\cite{Mikaelian72,Haeberl94}, that will become real after scattering on the grating. 
-         The expression for the intensity of this model is given in reference~\cite{p041}.
-        
-                        \begin{equation}\label{eq:BDR}
+                For a large number of periods one can integrate Eq.~\ref{eq:RDR_model} over the frequencies and obtain an analytical expression~(see paper~\cite{p021}) for the intensity of the SPR.
+        
+        \item
+                The model so-called Resonant Reflection Radiation (RRR) model based on the fact that a field of a moving charged particle could be described as a sum of the virtual plain waves~\cite{Mikaelian72,Haeberl94}, that will become real after scattering on the grating. The expression for the intensity of this model is given in reference~\cite{p041}.
+        
+                \begin{equation}\label{eq:BDR}
                         \begin{split}
                         &\left(
@@ -226 +196 @@
                         }         
                         \end{split}
-                        \end{equation}
-        \begin{equation}
-        \begin{split}
-        I&=C_1'\left( (E_x^D)^2 + (E_z^D)^2\right)
-        \end{split}
-        \end{equation}  
-        Here,$E^D_Z,E_X^D$ are the $Z$ and $X$ components of the field on the detector, $I$ is the Intensity of the radiation, $X_T, Z_T$ are the $X, Z$ coordinates on the Fig.~\ref{fig:geom1}, $K_1$ is the modified Bessel function
-of the second order, $\chi$ equals $1$ on the grating and $0$ in the gap, $\mathcal{R}(X_T,Z_T,\theta,\phi)$ is the grating-detector distance.
-
- In reference~\cite{p041}, by assuming the distances from the grating to be infinite, the authors also derived the far-zone approximation of the RRR model.
- 
-%       Although authors of the~\cite{p041} chose geometry of a plain strips
-%        it could be easily changed into the geometry discussed in this paper (Fig.~\ref{fig:geom1}). 
+                \end{equation}
+                \begin{equation}
+                        \begin{split}
+                        I&=C_1'\left( (E_x^D)^2 + (E_z^D)^2\right)
+                        \end{split}
+                \end{equation}  
+                Here,$E^D_Z,E_X^D$ are the $Z$ and $X$ components of the field on the detector, $I$ is the Intensity of the radiation, $X_T, Z_T$ are the $X, Z$ coordinates on the Fig.~\ref{fig:geom1}, $K_1$ is the modified Bessel function
+                of the second order, $\chi$ equals $1$ on the grating and $0$ in the gap, $\mathcal{R}(X_T,Z_T,\theta,\phi)$ is the grating-detector distance.
+
+                In reference~\cite{p041}, by assuming the distances from the grating to be infinite, the authors also derived the far-zone approximation of the RRR model.
         \end{itemize}
-
-
-         \section{Simulation of SEY for different models}
-         The parameters of the SPESO at SOLEIL synchrotron and E203 at FACET at SLAC experiments~{\cite{SPESO,p046}} were used in the simulation (see table~\ref{tab:SPESO_E203}). The constant of the RRR model was calculated from the assumption, that the intensities of the SC and RRR models are equal at $\theta=90^\circ$.
+        
+%=======================================================================================================
+\section{Simulation of SEY for different models}
+        The parameters of the SPESO at SOLEIL synchrotron and E203 at FACET at SLAC experiments~{\cite{SPESO,p046}} were used in the simulation (see table~\ref{tab:SPESO_E203}). The constant of the RRR model was calculated from the assumption, that the intensities of the SC and RRR models are equal at $\theta=90^\circ$.
 
         \begin{table}[!ht]
@@ -267 +234 @@
                 \normalsize
         \end{table}
-         \begin{figure}[!ht]
+        \begin{figure}[!ht]
                         \centering
                  \begin{subfigure}[t]{0.5\textwidth}
@@ -281 +248 @@
                         \caption{Calculated curves for the RDR (solid blue line), RRR (green line with circle marker), SC (blue dashed line) and GFW (purple line with square marker) models and their ratios.}
                         \label{fig:theta_RDR_SC_RRR}
-         \end{figure}
-%        \begin{figure}[!ht]
-%                       \centering
-%                       \includegraphics[width=0.5\textwidth]{MOPMB004f3.png}
-%                       \caption{Calculated curves for the RDR, RRR, SC and GFW models. Top plot is the calculated data, bottom plot is the ratio between models}
-%                       \label{fig:E203_theta_RDR_SC_RRR}
-%        \end{figure}
+        \end{figure}
                  
-         Taking into account an angular aperture of the detectors of 10$^\circ$, for each value of $\theta$ the intensity was integrated in $\phi$ over the range ${-5^\circ<\phi<5^\circ}$, in theta over the range ${\theta_i-5^\circ<\theta<\theta_i+5^\circ}$, where $\theta_i$ is the measurement angle. The calculation were done for ${40^\circ~<~\theta_i~<~140^\circ}$, with the step of $10^\circ$. 
+        Taking into account an angular aperture of the detectors of 10$^\circ$, for each value of $\theta$ the intensity was integrated in $\phi$ over the range ${-5^\circ<\phi<5^\circ}$, in theta over the range ${\theta_i-5^\circ<\theta<\theta_i+5^\circ}$, where $\theta_i$ is the measurement angle. The calculation were done for ${40^\circ~<~\theta_i~<~140^\circ}$, with the step of $10^\circ$. 
          
-         The figures~\ref{fig:SPESO_RDR_SC_RRR},~\ref{fig:E203_RDR_SC_RRR} show the comparison of the RDR, SC, RRR in the far zone, and GFW models, and their ratio. It is seen that for the RDR, SC and RRR models the difference is not greater than a factor of 2, which is within experimental errors. The GFW model gives intensity 10 times bigger, than the RDR and SC models, which could be explained by the fact, that in GFW calculations authors take into account the width of the grating, and the grating efficiency parameter is calculated numerically, for the case of N grating facets.
-         
-%        The figure~\ref{fig:SPESO_theta_RDR_SC_RRR} additionally has curve of the RRR model in the far-zone, normalized at $\theta=90^\circ$, and below the main plot is the ratio of the RRR and SC model, the ratio is not bigger than one order and have oscillations similar to the sine.
-         
-%        The figure~\ref{fig:far_correction} shows the correction factor, i.e. the ratio between the intensity of the SPR in the pre-wave zone and in the far-zone. The solid red line was made by calculating the ratio between RRR model and RRR model in the pre-wave zone.
-%        The blue dashed line is the correction factor calculated by considering the strips of the grating as oscillators, and then calculating interference in the pre-wave zone. Their difference is within 10\%, so it could be said that they are in agreement.
-         
-%       \begin{figure}[!ht]      
-%                       \centering
-%                       \includegraphics[width=0.45\textwidth]{MOPMB004f3.png}
-%                       \caption{Calculated curves for the RDR~(solid blue line) and SC~(dashed blue line) models. Top plot is the calculated data, bottom plot is the ratio between SC and RDR models}
-%                       \label{fig:SPESO_theta_RDR_SC}
-%        \end{figure}
-
-%        \begin{figure}[!ht]
-%               \centering
-%               \includegraphics[width=0.45\textwidth]{MOPMB004f5.png}
-%               \caption{The correction factor from the RRR~(solid red line) model and from the SC~(dashed blue line) model}
-%               \label{fig:far_correction}
-%         \end{figure}  
-
-%=======================================================================================================
-
-%       \subsection{Coherent Smith-Purcell Radiation}
-%       SPR models usually given for a single charged particle, because
-%        in this case it is possible to derive analytical solutions.
-%        To get the picture of the SPR intensity for a bunch, one need to use simple equation:
-%       \begin{equation}\label{eq:CohIncoh}
-%       \left( \frac{dI}{d\Omega} \right)_{N_e}=\left( \frac{dI}{d\Omega} \right)_1\left( N_e S_{inc} + N_e^2 S_{coh} \right)
-%       \end{equation}  
-%       Which is relation between the intensity per solid angle for a bunch
-%        $ \left( \frac{dI}{d\Omega} \right)_{N_e} $ and for a single electron $ \left( \frac{dI}{d\Omega} \right)_1 $,
-%        where $N_e$ is the number of electrons, $ S_{inc} $ and $ S_{coh} $ are coefficients corresponding
-%        to the incoherent and coherent cases, they are the Fourier transformations of the bunch profile.
-%        More detailed explanations are presented in the~\cite{p019}.
-         
-%       \section{Near-field zone effect}
-%               In every radiative phenomenon it is possible to identify the three zones~\cite{p013}: 
-%               \begin{enumerate}
-%               \item The wave-zone~--- at distances comparable to the wavelength,
-%               \item The far-zone~--- for the large distances at which the grating could be considered a single-point oscillator. In this zone intensity per solid angle is independent from the grating-detector
-%               separation.
-%               \item The pre-wave zone~--- between the first two zones, where the grating sizes should be taken into account. Here, intensity per solid angle is dependant on the grating-detector separation due to the
-%               interference effects.
-%               \end{enumerate}
-%               The criterion for far/pre-wave zones separation were calculated in  \cite{p041}. The condition for the far
-%               zone is:
-%                \begin{equation}
-%                \mathcal{R} \gg \frac{1}{n}dN^2(1+\cos{\theta}).
-%                \end{equation}
-%               For $N=800$ and $d=$\SI{0.05}{mm} at $\theta=90^\circ$, the far zone should be considered starting \SI{30}{m} which is much greater than available distances at the experiments, for other angles the far zone criterion is presented on the Fig.~\ref{fig:zones}, the (0,~0) coordinate correspond to the position of the grating.
-%               
-%       \begin{figure}[!ht]
-%               \centering
-%               \includegraphics[width=0.4\textwidth]{MOPMB004f2.png}
-%               \caption{Visualisation of the far/pre-wave zones}
-%               \label{fig:zones}
-%        \end{figure}
-                
-         
-         \section{Conclusions}
+        The figures~\ref{fig:SPESO_RDR_SC_RRR},~\ref{fig:E203_RDR_SC_RRR} show the comparison of the RDR, SC, RRR in the far zone, and GFW models, and their ratio. It is seen that for the RDR, SC and RRR models the difference is not greater than a factor of 2, which is within experimental errors. The GFW model gives intensity 10 times bigger, than the RDR and SC models, which could be explained by the fact, that in GFW calculations authors take into account the width of the grating, and the grating efficiency parameter is calculated numerically, for the case of N grating facets.
+
+%=======================================================================================================         
+\section{Conclusions}
         The SEY of the several leading models of the SPR were compared. The simulation shows that the SC and RDR models are in agreement within experimental errors. The RRR model is also close to the RDR and SC, but more detailed explanation on the constant is required.  More detailed consideration of the grating profile and width in the GFW simulation gives the intensity 10 times bigger. The ratios between the models are not changing much with the parameters~(except for the observation angle). This work will allow us to estimate the error due to theoretical uncertainty when SPR is used for longitudinal profile reconstruction.
-        % The calculations were also done for the E203 experiment~\cite{p046} at FACET at SLAC, and the  conclusions were similar.
-        
-%       While analysing the results of the SPR experiments, one should be aware of the pre-wave zone correction, that could be calculated using two approaches (RRR model and osillators approximation), that are giving close result.
-        
-        
-        %----------------------------------------------------------------------------------------
-        %       REFERENCE LIST
-        %----------------------------------------------------------------------------------------
+        
+%----------------------------------------------------------------------------------------
+%       REFERENCE LIST
+%----------------------------------------------------------------------------------------
+
 \begin{thebibliography}{99} % Use for 10-99 references
         \bibitem{Fitch99}
@@ -378 +279 @@
                 \emph{Phys. Rev. E}, vol. 50,
                 pp. R4294--R4297, Dec. 1994.\\
-%       \bibitem{SP53}
-%               S.~J.~Smith and E.~M.~Purcell.,
-%               ``Visible Light from Localized  Surface Charges Moving across a Grating'',
-%               \emph{Phys. Rev.}, vol. 92,
-%               pp. 1069-–1069.,        1953. \\        
         \bibitem{Nguyen97}
                 D.~C.~Nguyen,
@@ -454 +350 @@
                 San Sebastian, Spain, Sept. 2011,
                 paper MOP057, pp. 567--569.\\
-                                        
-                
-%       \bibitem{FELEvt},
-%               P.~Evtushenko \emph{et al.},
-%               ``Bunch length measurements at JLAB FEL'',
-%               in \textit{Proc. FEL 2006},
-%               BESSY, Berlin, Germany, Aug.--Sept. 2006,
-%               paper THPPH064, pp. 736--739.\\
-%       \bibitem{PLASMABerry06}
-%               F.-J.~Decker \emph{et al.},
-%               ``Multi-GeV Plasma Wakefield Accelera-tion Experiments'',
-%               In: \textit{E-167 Proposal},  2005,
-%               \url{https://www.slac.stanford.edu/grp/rd/epac/Proposal/E167.pdf}\\
-%       \bibitem{EOBerden2004}
-%               G.~Berden \emph{et al.},
-%               ``Electro-Optic Technique with Improved Time Resolution for Real-Time, Non destructive, Single-Shot     Measurements of Femtosecond Electron Bunch Profiles'',
-%               \emph{Phys. Rev. Lett.}, vol. 93,
-%               p. 114802.,  Sept. 2004. \\
-%       \bibitem{CTRMurokh98}
-%               A.~Murokh \emph{et al.},
-%               ``Bunch length measurement of picosecond electron beams from a photoinjector using coherent transition radiation'',
-%               \emph{Nucl. Instrum. Methods Phys. Res. Sect. A}, vol. 410, no. 3,
-%               pp. 452-–460, 1998 \\
-%
-%       \bibitem{SPExpWoods95}
-%               K.~J.~Woods \emph{et al.},
-%               ``Forward Directed Smith-Purcell Radiation from Relativistic Electrons'',
-%               \emph{Phys. Rev. Lett.}, vol. 74,
-%               pp. 1761–1764., Sept.   1992. \\
-%       \bibitem{p013}
-%               V.~A~Verzilov.,
-%               ``Transition radiation in the pre-wave zone'',
-%               \emph{Physics Letters A}, vol. 273, no. 1-2,
-%               pp. 135–140, 2000. \\
-%
-        
-
-
 \end{thebibliography}
-%\printbibliography     
+
 
 \end{document}