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\title{ Basic Physical Processes and Principles of Free Electron Lasers (FELs)}
\author{ Dagnachew W. Workie\\
         \\
         Department of Physics\\ 
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}
 
\date{November 26, 01 }
\maketitle
\begin{abstract}

 
A tunable coherent and high peak power sources have been very usefull in many applications. In this paper the physical process of  new type of coherent, flexible and very bright radiation source called Free-Electron Lasers $(FEL_{s})$ is descreibed. The motion of  electron beam in the presence of the optical and undulator fields leads to energy exchange. This energy exhange leads to  formation of bunches of electrons which will emit very bright radiation. The  resonance wavelength can,  in principle,be  tuned  to any radiation regions. This is possible by changing the undulatorparameter and/or electron beam energy. The line-shape of the emitted spectrum has the characteristic form of the short time emission process.  Different regions of the emitted radiation have found very wide applications.\\[1em]
 
\end{abstract}

\pagebreak

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\section{Introduction}



The Free electron lasers $(FEL_{s})$ was invented  by by J.M.J. Madey in 1971\cite{refE}. Free Electron Lasers are very flixible sources of coherent radiation. Due to their wide range tunabilty and high brghtness they have a growing applications. The $FEL_{s}$ system consists of an electron accelerator, an undulator in which the electrons emit the syncrotron radiation, and an optical resonator.
\section{Principles of operation of FEL}
In an FEL a beam of relativistic electrons produced by an accelerator passess through a transverse periodic field produced by a magnet called  undulator and exchanges energy with an electromagnetic field (Fig. 1).
\begin{figure}[htb]
\centerline{\psfig{figure=Fig1.ps,height=1.5in}}
\caption[]{\em{Basic components of FEL \cite{refC}.}}
\end{figure}
As a result of energy exchange, the electrons that gain energy begin to move ahead of the average electron, while the electrons that lose energy begin to fall behind the average .  The beam then  becomes bunched on the scale of the radiation wavelength and this collective motion of the bunches radiatespowerful coherent synchrotron radiation. Giga watts peak power have been demonstarted. The wavelength of the emitted radiation at the resonance  depends on the electron energy and the magnitude and periodicity of the undulator field according to the relation \cite{refA, refC}\
\begin{equation}\label{resonance}
\lambda_{o}=\frac{\lambda_{u}}{2\gamma^{2}}(1+\kappa^{2})
\end{equation}\
where $\lambda_{u}$ is the undulator period, $\gamma $ is the relativistic factor and $\kappa $ is the so called undulator parameter which is  proportional tothe magnetic field inside the undulator.\\

Typical values for electron beam enery ranges from few $MeV (\gamma\sim 1)$ toa few $GeV (\gamma\sim 1000)$; and the peak current varies from several ampersto many hundred of ampers. The peak undulator field strength $B_{o}$ is usually several KG ($B_{o}\sim2-7 KG$). The undulator period $\lambda_u$ is in the range of a few centimeters ($\lambda_u \sim 2-10cm $), whereas the undulator length $L_{u}=N\lambda_u$ extends for a few meters  ($L_{u}\sim 2-20 m$), so that the number of periods N can vary from 20 to several hundreds\cite{refA}.

\section{Basic Mechanism of FEL Interactions}
        

The motion of the  electron in the presence of the optical and the undulator field is governed by the Lorentz force equations \cite{refA, refC}:
\begin{equation}\label{motion1}
\frac{d}{dt}(\gamma \vec{\beta}) = -\frac{e}{mc}(\vec{E} + \vec{\beta}\times(\vec{B_u} +\vec{\beta}))
\end{equation}
\begin{equation}\label{motion2}
\frac{d}{dt}\gamma = -\frac{e}{mc}\vec{\beta}\cdot\vec{E}
\end{equation}
Where $\vec{E}$ and $\vec{B}$ are the electric and magnetic componenets of the e.m fields. Writting eqns.(\ref{motion1}) and (\ref{motion2}) interms of transverse and longitudinal components we obtain:
\begin{equation}\label{motion3}
\frac{d}{dt}{(\gamma \vec {\beta_{\perp}})} = \frac{e}{mc}{( \vec{\beta_{z}}\times\vec B_u)}-\frac{e}{mc}{(1-\beta_z)\vec{E}}
\end{equation}
\begin{equation}\label{motion4}
\frac{d}{dt}{[\gamma(1-\beta_z)]}=-\frac{e}{mc}{(\vec{z}\times\vec{B_u})\cdot \vec{\beta_{\perp}}}
\end{equation}
Where, $\vec{\beta_z} = {\frac{1}{c}\frac{dZ}{dt}}\hat{z}$. The first term on the right of eqn. (\ref{motion3}) accounts for the interaction of the electrons with the undulator magnetic field ; whereas  the the second term summerizes the effect of the transverse e.m fields,which combines almost to cancel out each other in the relativistic limit ($\beta_z \rightarrow 1$). Eqns. (\ref{motion3}) and (\ref{motion4}) desplay the complementary role of  the undulator and optical fields in  the FEL interaction.\\

        Considering the ideal plane -polarized undulator magnetic field,
\begin{displaymath}
\vec{B_u} = \hat{y}B_o \sin{k_uz},\qquad   k_u = \frac{2 \pi}{\lambda_u}
\end{displaymath}
and linearly ploarized emitted fields with a  complex amplitude $E(\vec{r},t)$ =$\mid{E{(\vec{r},t)}}\mid e^{i\phi{(\vec{r},t)}}$ , propagating along the z-direction,
\begin{equation}\label{field1}
\vec{E}(\vec{r},t) = ReE(\vec{r},t)e^{i(kz-\omega t)}\hat{x},\qquad \vec B(\vec{r},t) = \hat{z}\times\vec{E}
\end{equation}
eqns. (\ref{motion3})and (\ref{motion4}) becomes, respectively:
\begin{equation}\label{motion5}
\frac{d}{dt}(\gamma\beta_{\perp}) = -\hat{x}(\kappa\frac{d}{dt}\cos{k_u z} - K_R(\vec{r},t)[\frac{d}{dt}\sin\Psi(\vec{r},t)-\frac{d}{dt}cos\Psi(\vec{r},t)])
\end{equation}
\begin{equation}\label{motion6}
\frac{d}{dt}\gamma = \frac{\omega}{\gamma} K_{R}[\kappa\cos k_{u} z\cos\Psi(\vec{r},t) - K_R\sin\Psi(\vec{r},t)\cos\Psi(\vec{r},t)
\end{equation}
Assuming that the electron to experience a nearly constant  optical field (over $\lambda_u$) eqns ( \ref{motion5}) and (\ref{motion6}) can easily be integrated to get the longitudinal velocity $(\beta_z = [1-\beta_{\perp^2}-\frac{1}{\gamma^2}]^1/2)$\
\begin{equation}\label{beta1}
\beta_z = [ 1 - \frac{1}{\gamma^2}(1 + \kappa^{2}\cos^2{ k_u z} +K_R^2\sin^2{\Psi(\vec r,t)} - 2\kappa K_R\cos{k_uz}\sin{\Psi(\vec r,t)})]^{1/2}
\end{equation}\
where,\qquad
\begin{displaymath}
\kappa = \frac{eB_o}{mc\omega_u} ;\qquad and \qquad K_{R}(\vec{r},t) = \frac{e\mid E(\vec{r},t)\mid}{mc\omega},
\end{displaymath}
are the undulator parameter and the dimenssionless optical vector potential, respectively; and
\begin{displaymath}
\Psi{(\vec{r},t)} = (kz - \omega t) + \phi{(\vec{r},t)}
\end{displaymath}
\qquad is the phase angle combining of the fast and slow varing componnents of the e.m fields. Since usually  $\kappa > K_R $  by the order of magnitude, the  average value $\bar{\beta_{z}}(t)$ is given by:
\begin{equation}\label{beta2}
\bar{\beta_{z}}(t) = 1-\frac{1+\kappa^{2}/2}{2\gamma^{2}(t)}
\end{equation}\

On the other hand, requiring $\vec{\beta_{\perp}}$ to be in phase with the optical field during the motion assures that the electron transverse velocity retains its orentation relative to the optical electric field, thus simplifying that the energy exchange persists in the same direction. The resonant condition demands therefore  for  vanishing the difference between the optical phase and the electron transverse velocity phase at the instantaneous electron position,thus providing the expression\cite{refA}
\begin{equation}\label{condition1}
\omega - kc\bar{\beta_z} = \omega_u \bar{\beta_z}
\end{equation}\
For  propagation  in vacuum ($\omega = kc$ ) and using eqn. (\ref{beta2}) in the above relation one finds  the resonant frequency $\omega_o$ and wavelength $\lambda_0$ given by
\begin{equation}\label{condition2}
\omega_0 = \frac{2\gamma_0^{2}\omega_u}{1+\kappa^{2}/2},\qquad and \qquad \lambda_0 = \frac{\lambda_u}{2\gamma_0^2}{(1 + \frac{\kappa^2}{2})}
\end{equation}\

In general the electron phase in the combined optical and undulator fields is defined by\
\begin{equation}\label{condition3}
\xi(t) = kZ - (\omega-\omega_u)t = (k + k_u)\bar{Z} - \omega t
\end{equation}\
Introducing a dimensionless time\qquad $\tau =\frac{ct}{L_u}$,\qquad eqns. (\ref{beta2}) and (\ref{condition2}) gives us the  parameter measuring the degree of resonanace  (phase velocity):
\begin{equation}\label{velocity}
\upsilon(\tau)=\frac{d}{d\tau}{\xi(\tau)}=2\pi N [1- \frac{1+\kappa^{2}/2}{2\gamma^{2}(t)}]\frac{\omega_{0}(\tau)-\omega}{\omega_{0}(\tau)}
\end{equation}
where,\qquad $\omega_{0}(\tau)=2\frac{\gamma^{2}(\tau)\omega_u}{1+k^{2}/2}$ \qquad is the instantaneous resonance frequency.
       
\section{Spectral charactersitics  of the Emmitted Radaitions}


For relativistic electron moving on arbitrary curved trajectory, the electric field is  given by the standard expression\cite{refB}:
\begin{equation}\label{relative}
\vec{E(t)} = e[\frac{\hat{n}-\beta}{R^{2}\gamma^{2}(1-\hat{n}\cdot\beta)^3}]_{ret} +\frac{e}{c}[\frac{\hat{n}\times (\hat{n}-\beta)\times\dot{\beta}}{R(1-\hat{n}\cdot\beta)^3}]_{ret}
\end{equation}\
Where the subscript ret means that the quantieties in the square braket are evaluated at the retarded time $t^{'} = t - R(t^{'})$ and $R(t^{'})= \vec{X}-\vec{r}(t^{'})$ is the usual notation for observer position. Then the total energy detected during one particle passage per solid angle and per frequency, using the acceleration dependent electric field of eqn.(\ref{relative})is:\
\begin{equation}\label{spectra1}
\frac{d^{2}I}{d\Omega d\omega}=\frac{e^2}{4\pi^{2}c}\mid\int_{-\infty}^{\infty}\hat{n}\times (\hat{n}\times\beta)e^{i\omega (t^{'}-\frac{\hat{n}\cdot\vec{r(t^{'})}}{c})}dt^{'}\mid^2
\end{equation}\\
In evaluating the the spectral characteristic equation we will keep contributions upto the order $(\kappa/\gamma)^{2}$\cite{refE}. From Fig.2 and for $\psi\sim {1/\gamma}$,  $\hat{n}\cong (\psi\cos{\phi},\psi\sin{\phi}, 1-\frac{1}{2}\psi^2)$  then
\begin{displaymath}
[\hat{n}\times (\hat{n}\times\vec{\beta})]_{z}\cong\frac{\kappa}{\gamma}\psi\cos{\phi}\cos(\omega_u t)-\psi^2 ,\qquad [\hat{n}\times (\hat{n}\times\vec{\beta})]_{y}\cong\psi\sin{\phi},
\end{displaymath}\
\begin{equation}\label{approx1}
[\hat{n}\times (\hat{n}\times\vec{\beta})]_{x} \cong\psi\cos{\phi} + \frac{\kappa}{\gamma}cos(\omega_u t)
\end{equation}\

\begin{figure}[htb]
\centerline{\psfig{figure=Fig2.ps,height=1.5in}}
\caption[]{\em{Coordinates and definitions of angles \cite{refE}}}
\end{figure}\
The ossillating part in eqn.(\ref{spectra1}) can be written as:
\begin{equation}\label{approx2}
e^{i\omega(t-\frac{\hat{n}\cdot\vec{r}}{c})}\cong e^{i\frac{\omega}{2\gamma^2}(1+\frac{\kappa^{2}}{2}+\gamma^{2}\psi^{2})t}\sum_{m=-\infty}^{\infty}e^{-im\omega_{u} t}J_{m}(\zeta_{\omega},\xi_{\omega})
\end{equation}\
with\qquad $\zeta_{\omega}=-\frac{\kappa\omega}{\gamma\omega_{u}}\psi\cos{\phi},\qquad\xi_{\omega}=-\frac{\kappa^{2}}{8\gamma^{2}}\frac{\omega}{\omega_{u}}.$\\
We have denoted with $J_{m(x,y)}$ the generalized bessel function of first kind,whose generating function is provided by\
\begin{displaymath}
\sum_{m=-\infty}^{\infty}t^{m}J_{m(x,y)}=e^{[\frac{x}{2}(t-\frac{1}{t})+\frac{y}{2}(t^{2}-\frac{1}{t^2})]}.\
\end{displaymath}\
Using these resualts into our equation for spectral charactersitics (eqn.\ref{spectra1}) we finnally get:
\begin{equation}\label{spectra2}
\frac{d^2I}{d\Omega d\omega}=\frac{4e^2 N^2\gamma^4}{c}\sum_{m=1}^{\infty}\frac{m^2}{(1+\kappa^{2}+\gamma^2\psi^2)}[{\mid S_m^{(1)}\mid}^2 + {\mid S_m^{(2)}\mid}^2](\frac{\sin(\upsilon_{m}/2)}{\upsilon_{m}/2})^2
\end{equation}\
where \qquad
$\upsilon_m = 2\pi N(m-\frac{\omega}{\omega_1})$ ;  $\omega_1 = \frac{2\gamma^{2}\omega_u}{(1+\frac{\kappa^2}{2}+\gamma^2\psi^2)}$\
and
\begin{equation}
S_m^{(1)}=\psi\cos{\phi}J_m (\zeta_{\omega},\xi_{\omega})+\frac{\kappa}{2\gamma}[ J_{m-1}(\zeta_{\omega},\xi_{\omega})+J_{m+1}(\zeta_{\omega},\xi_{\omega})],
 S_m^{(2)}=\psi\sin{\phi}J_m(\zeta_{\omega},\xi_{\omega})
\end{equation}\
with the redefined quantities\
$\zeta_{\omega}=-m\frac{2\kappa\gamma\psi\cos{\phi}}{1+\frac{\kappa^2}{2}+\gamma^2\psi^2},\qquad \xi_{\omega}=-m\frac{\kappa^2}{4}\frac{1}{1+\frac{\kappa^2}{2}+\gamma^2\psi^2}$\\
        On the axis (i.e $\psi = 0$ or  $\xi = \xi_{\psi =0}$) eqn.(\ref{spectra2}) reduces to:\\
\begin{equation}\label{spectra3}
\frac{d^2I}{d\Omega d\omega}=\frac{e^2 N^2\gamma^2}{c}\sum_{m=1}^{\infty}F_{m}(\xi)[\frac{\sin(\upsilon_{m}/2)}{\upsilon_{m}/2}]^2
\end{equation}
where,
\begin{equation}
F_{m}(\xi)=(\frac{4m\xi }{\kappa})^{2}[J_{\frac{m-1}{2}}(m\xi)-J_{\frac{m+1}{2}}(m\xi)]^2
\end{equation}
Equations (\ref{spectra2}) and (\ref{spectra3}) shows that the line shape of  the emitted spectrum has the characteristic form of short-time emission process. The beam emittance and energy spread produce a broadening of the line spectrum and a reduction of the peak power\cite{refA,refC}.

\section{Tunability and Applicability of FEL in the Different Radiation Regions }

In section 2 we showed that the  Optical wavelength is  related to the the undulator period $\lambda_{u}$ and to the beam energies according to:
\begin{displaymath}
\lambda_0 = \frac{\lambda_u}{2\gamma_0^2}{(1 + \frac{\kappa^2}{2})},\qquad\kappa=\frac{eB_{o}\lambda_u}{2\pi mc^2}
\end{displaymath}\\
Tuning the operating wavelength is possible by adjusting the electron beam energy and /or the undulator  parameters.   In addition to its tunability $FEL_{s}$ high peak power and flexible pulse structure makes FEL a very attractive device, justified by the growing number of potential application depending on the region of radiation.\\

        The most compelling case for FLE facilitites starts with the far infrared region (1000 to $10 \mu m$). Commercial laboratory lasers are not effective in these region. This is the spectral region where molecule-surface vibrations, intermolecular cluster vibrations, etc. can be excited. Some of the areas wher this region is highly involved are: surface since, chemistry, solid state physics, Biophysics and plasma physics \cite{refE}. In Vacuum Ultraviolet and Extended Ultraviolet region (200 to 10 nm) laboratory lasers become increasingly inefficient. There are important scientific nead for these region, including photodissociation and single photon ionization detection.The X-ray region for the FLE (100 to 1 $A^{o}$). Because  there is no existing  x-ray FEL, studies involving coherent X-rays are totally unexplored. Applications of this region includes imaging, such as x-ray microscopy and holography \cite{refE}. 
      
\section{Conclusion}

Studing the interaction of a single particle in the presence of the undulator and optical fields gave us the very practical parameters of the  the Free Electron Lasers, i.e the tuning parameters. The tuning parameters are found to be the electron beam energy and/ or the undulator parameters.  In principle tuning of the $FEL_{s}$ to any radiation region is possible. The flexibility  of $FEL_{s}$ and its high peak power has found a grouing applications. Currently most of the $FELS_{s}$ are in the far infrared regions (1000 to 10 $\mu m$). However there is a high demand for the the Vacuum Ultraviolet and extended Ultraviolet rgion (200 to $10 nm$); and the X-ray region (100 to $1A^{o}$). 

\begin{thebibliography}{99}
\bibitem{refA} G. dattoli, A.Renieri and A.Torrer, Lecture on the Free-Electron Laser: Theory and and Related Topics, World Scientific (1993).
\bibitem{refB} J.D.Jackson, Classical Electrodynamics, Jhon Wiley and Sons Inc., (1975)
\bibitem{refC} Tmomas C.Marshall, Free-electron Lasers, McMillan publishing Company, NY (1985)
\bibitem{refD} S. Martellucci and Anthur N. Chester, Free electron Lasers, Plenum press, NY and London (1983)
\bibitem{refE} The National Academy of Sciences, Free electron lasers and Other Advanced Sources of Light: Scientific Research oppotunities, (1994)
\end{thebibliography}
\end{document}







