#LyX 1.3 created this file. For more info see http://www.lyx.org/ \lyxformat 221 \textclass book \begin_preamble \usepackage [] {dukedis} \author {Samadrita Roychowdhury} \title {High Brightness Electron And Photon Beams} \date {2006} \department {Physics} \advisor {V.N.Litvinenko} \mema {I.V.Pinayev,Co-Supervisor} \memb {S.Chandrasekhran} \memc {H.Gao} \memd {W.Tornow} \end_preamble \language english \inputencoding auto \fontscheme default \graphics default \paperfontsize default \spacing double \papersize Default \paperpackage a4 \use_geometry 0 \use_amsmath 0 \use_natbib 0 \use_numerical_citations 0 \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \quotes_times 2 \papercolumns 1 \papersides 1 \paperpagestyle default \layout Standard \begin_inset ERT status Open \layout Standard \backslash frontmatter \layout Standard \backslash copyrightpage \layout Standard \backslash makeabstract \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash newpage \layout Standard \backslash chapter{Abstract} \layout Standard Brightness is a practical measure of the ability of any beam to be focussed in all dimensions. Many applications, for example linear colliders in search of fundamental particles, and x ray light sources to name a couple, require high brightness electron beams. The state of the art in producing such high brightness electron beams is a radio frequency (RF) photoinjector. Photo-injectors are devices that combine the two well developed technologies of radio frequency guns with high power lasers. The distribution of the drive laser beam, used in a photo-injector, influences the brightness of electron beams. The brightness of an electron beam is diminished if the transverse and longitudinal distribution of the laser beam is non-uniform. In this thesis we describe the use of digital light processing (DLP) technique based on digital mirror device (DMD) for spatial modulation of the laser beam. A DMD is a micro electronic mechanical system (MEMS) comprising of millions of highly reflective micro mirrors controlled by underlying electronics. We discuss the new technique to measure QE of a photo-cathode using a DLP kit. We performed experiments at the Brookhaven National Laboratory, NY to investigate the effect of the spatial non-uniformity and modulation depth of the drive laser on electron beam brightness. We present results from these experiments. We also performed numerical simulations which agree with our experimental results and these results are also included in this thesis. \layout Standard \backslash newpage \layout Standard \backslash chapter*{Acknowledgements} \layout Standard First and foremost, I would like to thank my advisor Prof. Vladimir N. Litvinenko for his encouragement, support and mentoring of my graduate career. I feel fortunate for having had the opportunity to learn from his insight. It would be a daunting task to draw an education from such width and depth of knowledge but for his ever pervasive friendliness and willingness to teach. A very special thanks also goes to Prof. Igor V. Pinayev for his efforts and support in my education. Prof. Pinayev's ability to carry his expertise so lightly created an openness where one never felt the need to hold back a question as too dumb or an answer as too stupid. \layout Standard I would like to express my gratitude to Dr. Vitaly Yakimenko and the entire team at the Accelerator Test Facility at Brookhaven National Laboratory for giving me the opportunity to carry out my dissertation research at the ATF. Working at the ATF has been a time of much professional growth and I am thankful to all members of the staff who shared there expertise and knowledge so freely and amicably with me. I am especially thankful to Mr. Marcus Babzien for his proactive collaboration and the expertise that he provided to this project. I also thank Dr. Dmitry Kayran of the Collider Accelerator department at Brookhaven National laboratory for his patient support as I learned to navigate my way through quirks of PARMELA simulations. \layout Standard I express my gratefulness to Prof. Roxanne P. Springer for her support of my graduate studentship. But for the encouragement and support I have had from her, during some very difficult and adverse circumstances, the task at hand would have been overwhelming. \layout Standard I thank my friend and colleague Emily C. Longhi for sharing the best of times and helping me survive the worst moments of my years at Duke. I owe Emily much including perhaps my sanity during the final days of the preparation of this manuscript. I also thank my group mate Dr. Kevin Chalut for his cooperation and friendship. \layout Standard Though the past few years at graduate school have been a period of professional growth, I feel very thankful to my circle of friends who have made it a personally fulfilling venture. I cherish the friendships I have made at Duke, as much as the degree I receive. I especially acknowledge Dr. Suparna Kanjilal for her friendship and her ever willingness to lend an ear to my real and imagined woes. I also thank my friends Dr. Rukmini Kumar, Dr. Nirmala Vasudevan, Srividya Srinivasaraghvan and Sunethra Ramanan for the support I have had from them. \layout Standard I would like to thank my parents for their love, encouragement and support over all these years. I especially thank my mother for instilling in me a deep respect for learning and having faith in me and my abilities, particularly on occasions when I had none left. I thank my brother, Debjyoti, for setting the bar high at home and for being a constant source of encouragement. I thank my extended family, my sister in law Dola, my nephews Dhrupad and Sampad and my parents in law all of whom make an encompassing support system. \layout Standard Last but in no ways the least, I thank my husband Nilanjan who always rose to the occasion and turned himself deftly into a scientific sounding board, a matlab guru, and at times a punching bag for my failures; keeping me company when I needed it and putting up without mine as I disappeared for long periods to work in NY. I have little doubt, but for him, this manuscript would not have seen the light of day. \layout Standard \layout Standard \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash tableofcontents \layout Standard \backslash listoffigures \layout Standard \backslash listoftables \layout Standard \backslash dedication {to my mother...} \layout Standard \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash mainmatter \end_inset \layout Chapter Introduction \layout Standard \begin_inset ERT status Open \layout Standard \backslash pagenumbering{arabic} \end_inset \layout Quote \paragraph_spacing double \begin_inset Quotes eld \end_inset Everything should be made as simple as possible, but not simpler \begin_inset Quotes erd \end_inset \layout Quote \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ \SpecialChar ~ Einstein \layout Section \paragraph_spacing double Charged Particle Beams \layout Standard \paragraph_spacing double A charged particle beam is a group of particles that have nearly the same energy and move in about the same direction. Usually the total momentum along the beam trajectory is much higher than the transverse momentum of particles. The high energy and directionality of these particles make them useful for various applications. Though applications of particle beams usually conjure up images of large machines in high energy physics, particle beams have been continually expanding applications in many branches of research and technology. Such applications include flat screen cathode ray tubes, beam lithography for micro-circuits, thin film technology, and even the possible killing of anthrax spores at the United States postal service department \begin_inset LatexCommand \cite{key-83} \end_inset . \layout Standard \paragraph_spacing double A charged particle beam is primarily characterised by the type of particles, for example electrons, protons or alpha particles. Though further categories are often made in terms of many other parameters. To list a few : \layout Itemize \paragraph_spacing double the energy of the particles expresses in kilo-electron volts (KeV) or mega-elect ron volts (MeV) \layout Itemize \paragraph_spacing double the particle current expressed in Amperes (A) \layout Itemize \paragraph_spacing double the diameter of the beam \layout Itemize \paragraph_spacing double emittance, which is a measure of the phase space occupied by the beam. \layout Standard \paragraph_spacing double Of all charged particle beams, the technologically most important, is perhaps the electron beam. Electron beams find applications from the ubiquitous cathode ray tubes in televisions to sophisticated lithography techniques. It is also one of the most widely used probes to study the structure of material (electron microscope), and to explore the fundamental structure of particles ( \begin_inset Formula $e^{-}e^{+}$ \end_inset , \begin_inset Formula $e^{-}p$ \end_inset colliders). The quality of X-ray production has also improved since the introduction of electron accelerators. Electron accelerators have made possible entirely new kinds of light sources such as free electron lasers. The performance of all these applications depends on the quality of electron beams. It is due to advances made in producing better quality electron beams that the world's first X-ray free electron laser is around the corner. \layout Section \paragraph_spacing double Linac Coherent Light Source \layout Standard \paragraph_spacing double Linac Coherent Light Source (LCLS) is slated to become the first X-ray laser in the world, when it becomes operational in 2009. It is expected to produce X-rays much brighter and simultaneously much shorter than any present X-ray sources. The LCLS is different from any other X-ray source as it will provide laser like coherence and brightness to its pulses. It is also different from any existing laser source as its wavelength is in the X-ray part of the electro-magnetic spectrum and that has not yet been achieved by conventional lasers. This will open up exceptional opportunities to further our understanding of both the physical and the biological world. \layout Subsection \paragraph_spacing double LCLS - New Frontiers in Science \layout Standard \paragraph_spacing double We know that it is impossible to \begin_inset Quotes eld \end_inset see \begin_inset Quotes erd \end_inset an object smaller than the wavelength of probing light shining on the object. Thus the shorter the wavelength of light used, the better resolution we have over the details. Visible light at scale of hundreds of nano-meters is too large to be able to resolve atoms and single molecules. But X-rays with their wavelength in the range of few Angstroms can probe the atomic scales very well. In fact X-rays are already widely used to study and comprehend atomic arrangeme nts in various kinds of natural as well as man-made materials. Similarly the structural analysis of proteins and nucleic acid (DNA and RNA) has been made possible by bouncing off x rays from crystalline forms of these macro-molecules \begin_inset LatexCommand \cite{key-61} \end_inset . \layout Standard \paragraph_spacing double Presently scientists use X-rays produced by synchrotron light sources to study properties of materials as well as biological molecules. Though synchrotron light sources provide invaluable insight into the ultra-smal l world, the X-ray pulses created by these sources are not short enough to be able to capture that ultra-fast motion occurring in the realm of the ultra-small blocks of matter like nano-sized structures, or atoms and molecules. Atomic motion for example occurs at a time scale of less than trillionth of a second. Until now, what we can see, using X-ray beam from synchrotron sources, is the long exposure shots that give an average image of these constantly moving objects. The LCLS will emit X-ray pulses which are a factor of 1000 or more shorter than currently available X-ray pulses. These pulses will also be billion times brighter than any present X-ray source. The brighter the source, higher the number of photons, the clearer is the image. These exceptionally bright, coherent, short pulses of X-ray light holds the promise to give us a glimpse of the ultra-fast ultra-short world as has never been seen before. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/intro3.eps clip rotateOrigin centerBaseline \end_inset \layout Caption Science at LCLS. \begin_inset LatexCommand \label{cap:Science-at-LCLS.} \end_inset \layout Standard (a) Chemical phenomenon like molecular bond breaking will be filmed by LCLS pulses. (b) LCLS will take diffraction patterns of molecules at different positions within very short intervals. The resulting three dimensional reconstruction will reveal protein structures that cannot be crystallised. \layout Standard \shape italic \size footnotesize picture reproduced from http://www-ssrl.slac.Stanford.edu/lcls/index.html \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The applications of these pulses will come from a plethora of scientific fields. For example chemical reactions between small molecules are by nature ultra-fast. LCLS will provide a tool to record time -resolved images of chemical reactions. That would give scientists an opportunity, for the first time ever, to observe the making and breaking of bonds as the clouds of electrons that hold atoms together, shift. That is to akin to creating a motion picture using X-rays in chemical dynamics. Apart from enhancing our fundamental understanding of chemistry, this will lead to better understanding of natural chemical reactions like photo-synthesis which has major implications for future energy sources and agriculture. Another very active field of study that would benefit tremendously from the LCLS X-ray pulses is structural biology. The current tool of X-ray diffraction crystallography to study the structure of biological macro-molecules has its limitations. First, this technique is based on crystallography and an entire set of macro-molecules, those occurring in membranes, are very resistant to crystalliz ation. This resistance to forming crystals makes X-ray diffraction crystallography ineffective and the structure of these membrane based bio-molecules remains inaccessible. The LCLS pulses will be able to gather information about these structures without having to crystallize them. The pulses will be so short that it would be possible to record structure before the sample is destroyed as is the case with present day light sources. An understanding of such membrane bound proteins is an important goal for pharmaceutical industry. The access to the structure of these proteins would make it possible to design the drug molecules that target these proteins. \layout Standard \paragraph_spacing double The applications described above are by no means an exhaustive list of the utility of the ultra-fast high brightness pulses expected from the LCLS. The excitement runs across an entire gamut of scientific fields, from nano scientists waiting to understand the dynamics at those scales to astro-physicis ts watching out for atoms deluged by LCLS X-rays to reach states found in super planets like Jupiter and never before observed on earth. \layout Subsection \paragraph_spacing double LCLS-Free Electron Laser \layout Standard \paragraph_spacing double The LCLS is a special kind of laser. It is located at the powerful linear accelerator (linac) at Stanford Linear Accelerator Center (SLAC). The linac accelerates free electrons, that is electrons stripped off from atoms, to speeds comparable to the speed of light. These electrons then wiggle through an undulator magnet. Electrons being charged particles emit radiation every time they change direction. This emitted radiation can be in the X-ray range of the electro-magnetic spectrum by proper design of the source. This is a well known technique and is used in synchrotron light sources around the world, to produce X-rays. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Intro1.eps clip \end_inset \layout Caption Free electron laser scheme. \begin_inset LatexCommand \label{FELSchematic} \end_inset \layout Standard In a free electron laser, accelerated electrons follow an undulating path to emit radiation. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The LCLS will be different as it would provide hitherto unprecedented laser like qualities to the emitted X-rays. These X-rays will be coherent, the single most distinguishing feature of a laser light. Coherent means that all photons are in phase with each other and going in the same direction. To make these X-rays coherent the LCLS is operated as a free electron laser. The extra trick in such a laser is to make the electrons interact with the radiation they emitted zigzagging across the undulator. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/intro2.eps clip \end_inset \layout Caption Micro-bunching in free electron lasers. \begin_inset LatexCommand \label{MicroBunching} \end_inset \layout Standard (a) Electrons exchange energy with the co-propagating optical pulse.(b) Micro-bun ches of electrons are formed as a result of this exchange of energy. \layout Standard \size footnotesize \emph on picture reproduced from http://xfelinfo.desy.de/en/lichtquelle/1/index_noflash.html \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double At the start of the undulator, all the electrons have same energy. This changes, as the electrons interact with the emitted radiation co-propagati ng with the electrons down the undulator. The electrons can either absorb or relinquish energy to this emitted radiation. The absorption or release of energy causes the electron to speed up or fall behind the radiation. In an FEL, the magnetic structure and the speed of the electrons is coordinated such that the electrons fall behind the emitted radiation by exactly one wavelength after two crests (or troughs) of the undulator. This reiterates the release or absorption of energy, i.e. the electrons that lost some energy before, lose more energy and those that gained energy continue to do so. In this way electrons move close together into small groups. Thus the radiating light has now organised the electrons into small bunches each one optical wavelength away from each other. These small bunches of electrons are called micro-bunches. They continue to emit radiation but now the result is much more intense light as the radiation of the electrons in the micro-bunches, is in phase, and is thus coherent. A more detailed analysis of the theory of free electron lasers can be found in chapter 7. \layout Standard \paragraph_spacing double Critical to the success of the X-ray free electrons like LCLS \begin_inset Foot collapsed false \layout Standard The other X-ray free electron laser in design and construction phase is at Deutsches Elecktronen Synchrotron(DESY) laboratory in Hamburg,Germany. \end_inset is the quality of electron beams that have to be sent into the undulator. These ultra-short wavelength FELs impose severe requirements on the electron beams. The electron beam emittance \begin_inset Formula $\epsilon$ \end_inset [ \begin_inset LatexCommand \ref{Emittance} \end_inset ] must be much less than the wavelength of the free electron laser. \begin_inset Formula \begin{equation} \epsilon\ll\frac{\lambda_{fel}}{4\pi}.\label{eq:Ch1_emittance}\end{equation} \end_inset To produce electron beams that would measure up to fulfill the demands of X-ray free electrons lasers is one of the most challenging tasks for accelerato r scientists. The primary motivation for the research project, reported in this thesis, is to study and comprehend ways to better the quality of presently available electron beams. These free electron lasers also need very high peak currents of the order of magnitude of few Kilo-Amperes. \layout Section \paragraph_spacing double Electron Beams \layout Standard \paragraph_spacing double Before we take a look at what is meant by quality of a beam in the context of accelerators, let us review the physics behind the stability of so many charged particles, so close to each other. Charged particles in close proximity to each other are under the influence of strong repelling electromagnetic forces. These forces are called space-charge forces. From Coulomb's law we expect these charged particles to quickly diverge, blowing up the beam. But this situation becomes significantly different when all these charged particles propagate in the same direction at very high speeds. Particle beams with speed high enough to be comparable to the speed of light is called a relativistic beam. It is only because of the relativistic speeds that the electrons can form stable beams. A calculation of the forces that the charged particles experience in a relativistic beam makes it clearer. \layout Standard \paragraph_spacing double Assume a continuous stream of charged particles (say, electrons) moving along the z axis with velocity \begin_inset Formula $v_{z}$ \end_inset . These electrons will experience Lorentz force given by \begin_inset Formula \begin{equation} \vec{F}=e(\vec{E}+\vec{v}\times\vec{B})\end{equation} \end_inset \layout Standard \paragraph_spacing double Since physical phenomenon is independent of frame of reference \begin_inset Foot collapsed false \layout Standard The first postulate of relativity states that laws of nature and the results of all experiments performed in a given frame of reference are independent of translational motion of the system as whole. \end_inset , let us calculate the force in \begin_inset Formula $S^{*}$ \end_inset , a frame where the electrons are at rest. In such a frame the only force is the repelling electro-static force since the electrons are at rest \begin_inset Formula $(\vec{v}=0)$ \end_inset . The radial electric field \begin_inset Formula $E_{r}$ \end_inset at a distance \begin_inset Formula $r$ \end_inset from the beam axis can be derived from Coulomb's law, in cylindrical coordinate s, \begin_inset Formula $\bigtriangledown\cdot\vec{E}=4\pi\rho_{o}$ \end_inset which on integration gives \begin_inset Formula \begin{equation} \vec{E_{r}}=2\pi\rho_{0}r\hat{e_{r}}\end{equation} \end_inset \layout Standard \paragraph_spacing double Thus the magnitude of the force is \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} F^{*}=eE^{*}=e2\pi\rho_{o}^{*}r^{*}\end{equation} \end_inset \layout Standard \paragraph_spacing double Transforming the equations back to the laboratory system \begin_inset Formula \begin{equation} F_{r}=\frac{dp_{r}}{dt}\end{equation} \end_inset \layout Standard \paragraph_spacing double The radii and radial momentum remains the same in both reference frames, though the time and charge density in the laboratory frame are different from the rest frame. These relations can be expressed as \begin_inset LatexCommand \cite{key-62} \end_inset \begin_inset Formula \begin{eqnarray} r & = & r^{*}\\ p_{r} & = & p_{r}^{*}\\ \rho & = & \gamma\rho^{*}\\ dt & = & \gamma dt^{*}\end{eqnarray} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $\gamma$ \end_inset is the relativistic factor, whose values determine the measurable effects of relativity. \begin_inset Formula \begin{equation} \gamma=\frac{1}{\sqrt{1-(\frac{v}{c})^{2}}}\end{equation} \end_inset \layout Standard \paragraph_spacing double Thus the Lorentz force in the laboratory frame of reference becomes \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} F_{r}=2\pi e\frac{\rho_{o}}{\gamma^{2}}r\end{equation} \end_inset \layout Standard \paragraph_spacing double Thus we see that more relativistic the beam, that is closer the particle velocity is to speed of light (c), higher is the value of \begin_inset Formula $\gamma$ \end_inset , lesser is the effect of the repelling forces acting on the beam. Relativistic electron beams stabilise under the influence of their own fields. These beams must be accelerated to relativistic speeds rather quickly else the the repelling forces are strong enough to destroy the beam. RF-photo-injectors are presently the cutting edge in creating and maintaining such beams. \layout Subsection \paragraph_spacing double LCLS - Electron Beam \layout Standard \paragraph_spacing double The required parameters of the LCLS are listed in Table 1.1 \begin_inset LatexCommand \cite{key-63} \end_inset . Let us take a more detailed look at what these parameters mean and the state of the art in the reaching these parameters. The high energy that the electron beam must be accelerated to is made possible by accelerating these beams over large distances. The parameters of number of particles, bunch length and transverse emittance can be clubbed together to define a practical figure of merit for the electron beams. This figure of merit is called peak brightness. This brightness of the electron beam can be expressed as \layout Standard \paragraph_spacing double \begin_inset Formula \[ \textnormal{Brightness }\sim\frac{\textnormal{Number of Particles}}{\textnormal{BunchLength}\times\textnormal{Transverse Emittance}}\] \end_inset \layout Standard \begin_inset Float table wide false collapsed false \layout Caption LCLS Beam Parameters \begin_inset LatexCommand \label{cap:LCLS-Beam-Parameters} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard Energy [GeV] \end_inset \begin_inset Text \layout Standard 15 \end_inset \begin_inset Text \layout Standard Peak Current [kA] \end_inset \begin_inset Text \layout Standard 5 \end_inset \begin_inset Text \layout Standard Charge [nC] \end_inset \begin_inset Text \layout Standard 1 \end_inset \begin_inset Text \layout Standard Bunch Length [ \begin_inset Formula $\mu m$ \end_inset ] \end_inset \begin_inset Text \layout Standard 15 \end_inset \begin_inset Text \layout Standard Relative Energy Spread \end_inset \begin_inset Text \layout Standard \begin_inset Formula $2\times10^{-4}$ \end_inset \end_inset \begin_inset Text \layout Standard Transverse Normalised Beam Emittance [mm-mrad] \end_inset \begin_inset Text \layout Standard 1 \end_inset \begin_inset Text \layout Standard Relativistic Factor \begin_inset Formula $\gamma$ \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula $3.10^{4}$ \end_inset \end_inset \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double We can summarise the needs of a machine like the LCLS by saying, that it needs a very high brightness electron beam. This high brightness electron beam requires more charge, short bunches, and small transverse emittance. Transverse emittance, is the finite phase space volume i.e the product of the size and the momentum spread. The ideal beam, of course, would have zero cross-sectional area and all the electrons would be headed in the same direction. A more detailed analysis of the concepts of brightness and emittance and can be found in chapter 4. \layout Section \paragraph_spacing double Motivation-Brighter Electron Beams \layout Standard \paragraph_spacing double The quality of the electron beams is determined largely by the electron sources. Electron sources in most day to day applications use electron thermionic emission, field emission, and photo-emission. Significant thermionic emission occurs only at relative high temperature ( \begin_inset Formula $1000^{o}C$ \end_inset ), the thermionic current density is low and and the bunch is long. These limitations make it unsuitable for serving as source for X-ray FELs. Another technique of producing electrons is field emission. Field emission is a form of quantum tunnelling, where electrons tunnel through a barrier in presence of very high electric fields [GeV/m]. The low current of the field emission sources is a major limitation. The state of the art for producing high brightness electron beams is the photo-cathode RF gun with photo-emission process initiated by a powerful drive laser. \layout Standard \paragraph_spacing double The photo-cathode RF gun combines the two well developed technologies of high power lasers and high gradient radio frequency linear accelerators. A photo-cathode is a conductive surface quoted with a photo-sensitive material. When a photo-cathode is struck by a high power laser light, electrons are ejected out of it due to photo-electric effect. Photo-emission gives a current density at least orders of magnitude greater than thermionic emission. These electrons are accelerated by a RF linac, whose primary goal is to accelerate electrons, at the expense of RF power, to speeds comparable to speed of light. If the RF field is strong enough to accelerate electrons to relativistic energies, before it can be affected by space charge forces, a very high brightness, up to 5x \begin_inset Formula $10^{15}$ \end_inset \begin_inset Formula $\mathrm{\mathrm{\mathrm{A.m^{-2}.rad^{-2}}/cm^{2}}}$ \end_inset \begin_inset LatexCommand \cite{key-64} \end_inset can be achieved. The term photo-injector is often used to refer to a system comprising of all the three parts: photo-cathode, high power laser, and the RF gun. A detailed review of the physics and technology of photo-injectors can be found in chapter 3. The primary motivation for the projects reported in this thesis (chapters 2 through 6), is to improve the brightness of the present day electron beams. \layout Section \paragraph_spacing double Towards Brighter Beams \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/intro4.eps clip rotateOrigin centerBaseline \end_inset \layout Caption Non-Uniform laser beam on the cathode \begin_inset LatexCommand \label{IntroNonUniform} \end_inset \layout Standard (a) Shows the image imposed on a digital mirror device ( \begin_inset LatexCommand \ref{dmd} \end_inset ) to distort the laser distribution incident on the photo-cathode.(b) Shows the image of the laser beam under such distortion. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The brightness of electron beams depend on the optimal performance of all components of the photo-injector. In this thesis we focus on the efficiency of the photo-cathode and the laser shape used to illuminate the photo-cathode. The performance of the drive laser determines the properties of the electron beam. The electron beam mimics the properties of the laser in terms of longitudinal and transverse distributions. These distributions may affect the emittance of a high brightness electron beam. The electron beam which develops a non-uniform distribution, as a consequence of the laser non-uniformity, will experience more emittance growth as compared to the electron beam that corresponds to a uniform laser beam distribution.. We conducted experiments at the Brookhaven Accelerator Test Facility (ATF) to study the emittance growth due to transversely non-uniform laser beams. We distort the laser beam using the digital light processing kit ( DLP \begin_inset Formula $^{TM}$ \end_inset ) and study the effect on the emittance. Fig 1.4 shows an example of one such controlled non-uniformity imposed upon the laser and the electron beam. We found that only macroscopic non-uniformity affects the emittance. We found very good agreement with the experimental results using numerical simulations done with PARMELA. We report experimental results and simulations in chapter 6. \layout Standard \paragraph_spacing double We also report on the novel technique developed to measure the quantum efficienc y (QE) of the photo-cathode at ATF. We developed the control system of the digital light processing device. This device is now used to scan over the surface of the ATF photo-cathode and can deliver a QE map of the surface in about twenty minutes. This is much faster than the traditional methods, and also provides much higher resolution over the QE maps. We report on the control system as well the results and insights obtained by measuring QE by this technique. Fig 1.5 shows one such QE map. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/intro5.eps clip \end_inset \layout Caption Quantum Efficiency map obtained by using digital light processing kit on ATF photo-cathode. \begin_inset LatexCommand \label{IntroQE} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard This technique has the potential to correct for the non-uniformities in electron beam, that arise out of the surface non-uniformities of the cathode. Most cathodes are not equally efficient over the entire surface area, this leads to non uniform emission and in turn adds to non-uniformity in the electron beam. The digital light processing technique can ameliorate this effect, by illuminat ing less efficient spots with higher intensity light. Such a compensation technique can lead to more uniform electron beams and would bring us a step closer to ultimate brightness in electron beams. \layout Chapter ATF Overview \begin_inset LatexCommand \label{ATFOverview} \end_inset \layout Section Introduction \layout Standard \paragraph_spacing double The Accelerator Test Facility (ATF) at the Brookhaven National Laboratory (BNL) is an advanced accelerator facility that provides high brightness electron beams and high power, short pulse laser beams for reserach in the physics of particles and laser beams. The core capabilities of the ATF comprise of a high brightness photo-injector electron gun, a 75 MeV Linac, high power lasers that are synchronised with the electron beam to the level of picoseconds and four beam lines. The photo-injector is a 1.5 cell photo-cathode RF gun with magnesium cathode, operating at 2856 MHz. The laser system includes a giga watt peak power, 1064 nm wavelength, 15 ps Nd:YAG laser and a gigawatt \begin_inset Formula $CO_{2}$ \end_inset laser which is at 10.6 \begin_inset Formula $\mu m$ \end_inset . The Nd:YAG laser is used to produce photoelectrons in the electron gun and simultaneously switch a short pulse from the \begin_inset Formula $CO_{2}$ \end_inset laser. A pulse from the Nd:YAG laser is split, a part of it is frequency quadrupled (266 nm UV) and sent to illuminate the cathode, the other part of the pulse is used to switch the \begin_inset Formula $CO_{2}$ \end_inset laser. The use of the same pulse to control both the linac and the \begin_inset Formula $CO_{2}$ \end_inset slicing system ensures synchronisation of the e-beam with the \begin_inset Formula $CO_{2}$ \end_inset laser pulse on picoseconds level. The electrons are accelerated up to 4.5 MeV energy at the end of the RF gun. Then the electrons are accelerated by two sections of the linac and can reach energies up to tens of MeV. The gun and the linac are powered by two separate klystrons. The phase and amplitude of these power sources can be independently adjusted by electronics phase shifters and variable attenuators respectively. A sophisticated computer control system controls the set point of these elements. The ATF can deliver electron beams with typically 1nC charge and 1 to 8 ps pulse length up to an energy of 70 MeV. \layout Section \paragraph_spacing double The Nd:YAG Laser System \begin_inset LatexCommand \label{optics} \end_inset \layout Standard \paragraph_spacing double The schematic diagram of the YAG system is shown in figure \begin_inset LatexCommand \ref{cap:Gun-Hutch-optics} \end_inset . The energy of the laser is about 30 mJ and the pulse length is 15 ps. The pulse from the YAG laser is split and about 60% of the initial laser energy is sent to the CO \begin_inset Formula $_{\textrm{2}}$ \end_inset system to produce slicing. The rest of the pulse is doubled and then quadrupled to get UV light at 266nm. This UV light travels across about 15 m of enclosed transport line to illuminat e the cathode of the RF gun. This UV pulse is about 8ps FWHM pulse length and about 0.2mJ of energy. Nearly one third of the energy can be made available, by focusing the beam, to a 3 mm beam size on the cathode. The optics are shown in figure \begin_inset LatexCommand \ref{cap:Gun-Hutch-optics} \end_inset . The beam is split before the cathode to form image of the cathode. This is done to collect information about various laser properties like intensity and beam size at the cathode. The various laser parameters such as power, and vertical or horizontal positions are under computer control and can be changed remotely. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/gun_hutch.eps display none rotateOrigin center \end_inset \layout Caption Gun Hutch optics \begin_inset LatexCommand \label{cap:Gun-Hutch-optics} \end_inset \layout Standard The UV light from the laser room travels for about 15 m in enclosed transport line. The optics provide steering control of the laser beam on the cathode, the time slew of the laser wavefront, beam energy diagnostics, beam shaping diagnostics and imaging of the cathode surface. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section \paragraph_spacing double The Electron RF GUN \layout Standard \paragraph_spacing double The electron source of the ATF is called a photo-cathode RF gun or photo-injecto r. This type of gun comprises of a photo-cathode located inside of an RF cavity. Electrons are generated by illuminating the photo-cathode by laser pulses. The generated electrons are made relativistic within very short distances by operating the RF cavity at very high accelerating fields. High surface field on cathode and the high yield of electrons due to photo-emis sion both lead to a very large current density. Since brightness is proportional to current density, high brightness beams can thus be achieved. \layout Standard \paragraph_spacing double The photo-injector at ATF, BNL is a resonant \begin_inset Formula $\pi$ \end_inset -mode 1.6 cell cavity operating at 2856 MHz. The metal photo-cathode is located on the back wall of the RF cavity. Cathodes can be changed by using the 'choke-joint' access port. High power microwaves (up to 6MW) drive the cavity to high electric fields. At the maximum peak power of 6.1 MW the peak surface electric field is 119 MV/m and the cathode field is 100MV/m. The primary motivation for such high fields is to minimise the time the electron bunch is affected by strong space charge forces. The high field leads to rapid acceleration, and the electron bunch attains relativistic speeds withing a short duration. Relativistic speeds minimise the effects of space charge as relativistic particles stabilise under self fields. The other advantage of having such high fields is that it increases the quantum efficiency of the cathode by Schottky effect \layout Standard \begin_inset Float table wide false collapsed false \layout Caption ATF gun Parameters \begin_inset LatexCommand \label{cap:ATF-Gun-Parameters} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard RF frequency (MHz) \end_inset \begin_inset Text \layout Standard 2856 \end_inset \begin_inset Text \layout Standard Cathode cell length (cm) \end_inset \begin_inset Text \layout Standard 2.625 \end_inset \begin_inset Text \layout Standard Second cell length \end_inset \begin_inset Text \layout Standard 5.23 \end_inset \begin_inset Text \layout Standard Cell diameter (cm) \end_inset \begin_inset Text \layout Standard 8.31 \end_inset \begin_inset Text \layout Standard Radius of aperture \end_inset \begin_inset Text \layout Standard 1.25 \end_inset \begin_inset Text \layout Standard Field on cathode (MV/m) \end_inset \begin_inset Text \layout Standard 100 \end_inset \begin_inset Text \layout Standard Peak field on wall (MV/m) \end_inset \begin_inset Text \layout Standard 119 \end_inset \begin_inset Text \layout Standard RF power (MW) \end_inset \begin_inset Text \layout Standard 5.9 \end_inset \begin_inset Text \layout Standard Cavity Q \end_inset \begin_inset Text \layout Standard 12000 \end_inset \begin_inset Text \layout Standard Effective shunt impedance (M \begin_inset Formula $\Omega$ \end_inset /m) \end_inset \begin_inset Text \layout Standard 57 \end_inset \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section \paragraph_spacing double LINAC And Beam Lines \layout Standard \paragraph_spacing double ATF Linear Accelerator is a traveling wave SLAC type accelerator which consists of two sections and three experiment beam lines. Each section of the LINAC is 3 m long and operates at 2856 MHz. The linac is driven by a high power microwave tube called klystron. The ATF klystron can go up to 28 MW in power. With this power the electron beam can be accelerated up to 68 MeV and thus obtain a total of up to 72 MeV (about 4.5 MeV comes from the gun). The set up comprises a gun, solenoid and monitor named LPOP-UP 1, which is also a Faraday cup. The Faraday cup is an isolated electrode which stops and captures the beam charge. An oscilloscope is used to measure the current across the Faraday cup when an electron bunch hits it. The oscilloscope software can read the area of the current signal, which is the charge from the radio frequency gun. Thus LPOP-UP1 is the monitor where charge is read. The ATF diagnostics also include beam profile and position monitors that produce image of the beam on a video monitor. This is achieved by converting beam impact to visible radiation (on a phosphor screen for example) and then using a CCD camera to image the optical radiation. A 'frame-grabber' that converts video signal to a digital matrix, captures the information from the camera. There are also vertical and horizontal steering magnets at the entrance and inside the linac to align the position and angle of the beam. Downstream of the linac, there are 3 beam lines comprising of quadrapole magnets, trim magnets, diagnostics and vacuum equipment which deliver the beam to various experimental locations. \layout Section \paragraph_spacing double Slice Emittance Measurement of Electron-Bunches at ATF \begin_inset LatexCommand \label{slice-emittance} \end_inset \layout Standard \paragraph_spacing double The emittance compensation scheme developed and tested at ATF provides a near complete characterization of the 6D phase-space electron density distribut ion. Figure \begin_inset LatexCommand \ref{Ch2_F2measure emittance} \end_inset gives the schematic layout of the slice emittance measurement diagnostic. The beam is generated at the photo-cathode by 10 ps long UV laser pulse. The electron bunch is quickly accelerated in the two cell RF cavity, and reaches an energy of about 4 MeV. The beam emerging from the gun is focused by a solenoid magnet and injected into a 2856 MHz linear accelerator. A laminar flow beam waist is formed in the linac. The acceleration continues till space charge forces are negligible and the phase space distribution is effectively frozen. The linac has two sections, and the phase of the second section can be controlled independently by a mechanical phase shifter. Detuning the phase of the second linac with respect to to the first creates a chirp in energy of the beam. The beam is then bent horizontally by a \begin_inset Formula $20^{o}$ \end_inset dipole magnet and a small slice in energy is selected by a slit. A quadrupole magnet, placed after the slit, is scanned in current. For each current value, the vertical beam size is measured on a beam profile monitor. The beam profile monitor (BPM) comprises a phosphorous screen to form an image of the beam and a CCD camera to record the image. Thus the second linac section, dipole and the slit form a filter selecting a slice of the beam with a particular energy and the quadrupole lenses and the beam profile monitor form the analyser to measure the beam matrix. \layout Standard \paragraph_spacing double If the coupling among the different dimensions is negligible, the particle distribution in the vertical plane is represented by the symplectic beam matrix. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/emittance_measure.eps scale 35 rotateOrigin leftBaseline \end_inset \layout Caption Slice emittance measurement schematic. \begin_inset LatexCommand \label{Ch2_F2measure emittance} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard Considering the space charge forces as negligible an equation can be written based on the Twiss parameters of the beam \begin_inset LatexCommand \cite{key-67} \end_inset \layout Standard \begin_inset Formula \begin{equation} \sigma_{2}^{2}=\epsilon\left(\beta-2\alpha_{1}L+L^{2}\gamma_{1}\right))-\frac{\epsilon}{f}\left(2L\beta_{1}-2L^{2}\alpha_{1}\right)+\frac{\epsilon}{f^{2}}\left(L^{2}\beta\right)\end{equation} \end_inset \layout Standard \paragraph_spacing double The procedure then is to measure \begin_inset Formula $\sigma^{2}$ \end_inset , i.e. the mean square beam size versus the focal length of the lens and fit the resulting curve to calculate emittance. The experimental data is obtained by the BPM. The image is captured by a CCD camera with a frame grabber. The CCD camera has pixel size of 11 x 13 microns and the number of pixels is 512 x 480. The digitised image is analysed for emittance. The analysis consists of integrating over the horizontal direction and obtaining the rms beam size for the vertical direction. Each vertical rms size corresponds to one point in the quadrapole scan. The error estimate in the emittance measurement is based on the resolution of the digital image. This resolution is determined by the combination of pixel size and spatial resolution of the phosphor. \layout Chapter Physics and technology of Photo-injectors \layout Section Introduction \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/BNL_gun.eps display none \end_inset \layout Caption BNL Photo-Cathode gun \begin_inset LatexCommand \label{cap:BNL-Photo-Cathode-Gun} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double Photo-Injectors are the state of the art for high brightness electron sources. Photo-Injectors combine the two well developed technologies of high power lasers and radio-frequency linear accelerators. The laser is used to produce and control the electron beam and a RF cavity is used to accelerate the electrons quickly to relativistic energies to minimise the effect of the self fields. In the latter half of the introduction we look at the source requirements for various applications that is demanded of photo-injectors. In section \begin_inset LatexCommand \ref{sec:Photo-Injector-Operational-Characteristics} \end_inset we look at operational characteristics of photo-injectors. This section covers three topics: photo-cathodes, lasers and the various guns employed in photo-injectors as each plays a significant role in production of high brightness electron beams. Section \begin_inset LatexCommand \ref{Emittance} \end_inset introduces the figures of merit associated with high brightness electron beams namely emittance and brightness. In section \begin_inset LatexCommand \ref{sec:Transport-of-beam} \end_inset we discuss beam transport and techniques to measure emittance. Finally in section \begin_inset LatexCommand \ref{sec:FEL-Applications} \end_inset we look at some applications of photo-injector technology with special focus on free electron lasers. \layout Standard \paragraph_spacing double Photo-injectors routinely generate current density greater than 500A/cm \begin_inset Formula $^{2}$ \end_inset . This current density is usually only limited by the field gradient on the cathode or the laser intensity. In terms of charge most photo-injectors generate between 1 and 10 nC per micro-pulse. Typical pulse lengths less than 10 ps are produced for charges of 1nC. Since a photo-injector is a source ( of electron beams) hence reliability and availability are primary concerns and they must be taken into account before selecting specifications. Depending on requirements such as polarized or unpolarized electrons, high charge per bunch or high average current, different setups may be selected. The photo emission threshold defines the laser wavelength the Quantum Efficienc y defines the laser output power and vacuum conditions and sensitivity to the electric field defines the gun. Table 3.1 shows some examples of source requirements for: Linear Colliders (LC) \begin_inset LatexCommand \cite{key-68} \end_inset , Self-Amplified Spontaneous Emission Free Electron Laser (SASE-FEL) \begin_inset LatexCommand \cite{key-68} \end_inset , Energy Recovery Linear Accelerators (ERL) \begin_inset LatexCommand \cite{key-70} \end_inset , Laser Wakefield Accelerators (LWA) \begin_inset LatexCommand \cite{key-71} \end_inset , and fourth generation X ray light sources (Greenfield) \begin_inset LatexCommand \cite{key-69} \end_inset \layout Standard \begin_inset Float table wide false collapsed false \layout Caption Various Source Requirements \begin_inset LatexCommand \label{cap:Various-Source-Requirements} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard \end_inset \begin_inset Text \layout Standard \begin_inset Formula $I(Amp)$ \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula $\tau_{FWHM}$ \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula $\epsilon_{n}(mm.mrad)$ \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula $B_{n}(Am^{-2}.rad^{2})$ \end_inset \end_inset \begin_inset Text \layout Standard LC \end_inset \begin_inset Text \layout Standard 500 \end_inset \begin_inset Text \layout Standard 8 \end_inset \begin_inset Text \layout Standard 10 \end_inset \begin_inset Text \layout Standard 1.10 \begin_inset Formula $^{\textrm{13}}$ \end_inset \end_inset \begin_inset Text \layout Standard SASE-FEL \end_inset \begin_inset Text \layout Standard 180 \end_inset \begin_inset Text \layout Standard 6 \end_inset \begin_inset Text \layout Standard 2 \end_inset \begin_inset Text \layout Standard 9.10 \begin_inset Formula $^{\textrm{13}}$ \end_inset \end_inset \begin_inset Text \layout Standard ERL \end_inset \begin_inset Text \layout Standard 50 \end_inset \begin_inset Text \layout Standard 3 \end_inset \begin_inset Text \layout Standard 1 \end_inset \begin_inset Text \layout Standard 10 \begin_inset Formula $^{\textrm{14}}$ \end_inset \end_inset \begin_inset Text \layout Standard LWA \end_inset \begin_inset Text \layout Standard 1000 \end_inset \begin_inset Text \layout Standard 0.2 \end_inset \begin_inset Text \layout Standard 3 \end_inset \begin_inset Text \layout Standard 2.10 \begin_inset Formula $^{\textrm{14}}$ \end_inset \end_inset \begin_inset Text \layout Standard GreenField \end_inset \begin_inset Text \layout Standard 500 \end_inset \begin_inset Text \layout Standard <1 \end_inset \begin_inset Text \layout Standard 0.1 \end_inset \begin_inset Text \layout Standard >10 \begin_inset Formula $^{\textrm{17}}$ \end_inset \end_inset \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section \paragraph_spacing double Photo-Injector Operational Characteristics \begin_inset LatexCommand \label{sec:Photo-Injector-Operational-Characteristics} \end_inset \layout Subsection \paragraph_spacing double Photo-cathodes \layout Standard \paragraph_spacing double One of the most important element which determines the reliability and stability of photo-injector is the photo-cathode. Robust high efficient photo-cathodes reduce the cost of the laser system and also improve the stability and reliability of the laser system by reducing the heat load of the laser system. Unfortunately there exists no photo-cathode versatile enough to fulfill all requirements and compromises must be made to suit particular specifications. We discuss three types of photo-cathodes which are representative of the R&D program in the field of photo-injectors. \layout Subsubsection Metallic Photo-cathodes \layout Standard \paragraph_spacing double These cathodes are very attractive as they are relatively easy to produce; they can be transported and installed in ambient air, and in principle have an infinite life time. These photo-cathodes also have added advantage of being able to tolerate high electric fields and poor vacuum conditions. In practise, however, many of the benefits are not realised as access to a clean metal surface is inhibited by contamination. These contaminants often, cannot be removed by simple chemical cleaning but require an in-situ treatment with laser light. This treatment needs to be repeated depending upon the vacuum conditions and the extracted charge. \layout Standard \paragraph_spacing double The other disadvantages of metal photo-cathodes arise due to high reflectivity of metals ( \begin_inset Formula $30$ \end_inset % to \begin_inset Formula $90$ \end_inset %) and high work function. The high reflectivity leads to only a small amount of photons being absorbed and the high work function ensures that only very energetic photons that can pass through the surface barrier (3-5eV) contribute to releasing photo-elec trons. Of the released photo-electrons only a small fraction (few %) reach the vacuum interface and the rest are lost to high electron electron scattering \begin_inset Foot collapsed false \layout Standard The principal energy loss mechanism in the metals is electron-electron scatterin g. Near threshold a primary electron can lose a significant fraction of its energy in a single scattering event, while the secondary electron may not gain enough to allow it to escape. \end_inset in the conduction band. This leads to low QE ( \begin_inset Formula $10^{-7}$ \end_inset to \begin_inset Formula $10^{-3}$ \end_inset ) and the use of UV light (200-300 nm). The low QE puts a limit on the laser energy density to be below the plasma generation level, which in turns limits the total emitted charge density to below a few \begin_inset Formula $nC/mm^{2}$ \end_inset for ps pulses. The emitted average current is limited to a few \begin_inset Formula $\mu A$ \end_inset by the available average laser power. \layout Subsection Alkali Photo-cathodes \layout Standard \paragraph_spacing double The \begin_inset Formula $Cs_{3}Sb$ \end_inset multi alkali cathode was the first cathode to be used in a photo-injector. Since then many compounds of the alkali-antimonide family \begin_inset Formula $($ \end_inset \begin_inset Formula $Cs_{3}Sb$ \end_inset , \begin_inset Formula $Na_{2}K(Cs)Sb$ \end_inset \begin_inset Formula $)$ \end_inset have been used successfully as photo-cathodes. These compounds work in the visible light making the drive laser requirements less stringent. Unfortunately since they are used with \emph on l \emph default ower \emph on \emph default energy photons they tend to be susceptible to contamination. \layout Standard Another family, alkali-tellurides ( \begin_inset Formula $Cs_{2}Te$ \end_inset , \begin_inset Formula $Rb_{2}Te$ \end_inset , \begin_inset Formula $RbCsTe$ \end_inset , \begin_inset Formula $K_{2}Te$ \end_inset ...) has been shown to be good candidates for photo-cathodes despite having larger photo-emission thresholds (3.5-4 eV) requiring UV light. These photo-cathodes can operate from weeks to months with a QE >1.5% and the mean current is at least 1 \begin_inset Formula $mA$ \end_inset (limited by the laser). These photo-cathodes were intensively used at DESY \begin_inset Foot collapsed false \layout Standard Deutsches Elektronen Synchrotron ("German Electron Synchrotron") is the biggest German research center for particle physics, with sites in Hamburg and Zeuthen. \end_inset in the Tesla Test facility-12 cathodes in 4 years and at CERN \begin_inset Foot collapsed false \layout Standard \begin_inset ERT status Open \layout Standard CERN is the European Organization for Nuclear Research (Organisation Europ \backslash 'eenne pour la Recherche Nucl \backslash 'eaire), is the world's largest particle physics laboratory, situated on the border between France and Switzerland. \end_inset \end_inset in the CLIC Test Facility -65cathodes in 10 years. \layout Standard \paragraph_spacing double The alkali photo-cathodes must be prepared in ultra high vacuum (UHV) environmen t and they have to be used without breaking the vacuum. This means the use of load-lock system and eventually an UHV transport carrier if th \emph on e \emph default preparation chamber is not attached to the photo-injector. The stringent vacuum conditions and the high cost of sophisticated fabrication required by these photo-cathodes often offset the advantages of high QE. \layout Subsection NEA Photo-cathodes \layout Standard \paragraph_spacing double Negative electron Affinity (NEAS) photo-cathodes are produced by co-adsorptions of electro positive elements like \begin_inset Formula $Cs$ \end_inset or a combination of \begin_inset Formula $Cs$ \end_inset and \begin_inset Formula $O$ \end_inset to reduce the work function such that the vacuum level lies below the lower edge of the conduction band. These cathodes are efficient photo-emitter in the visible and near infra -red region. Typical NEA semiconductors are \begin_inset Formula $Si$ \end_inset , \begin_inset Formula $GaAs$ \end_inset , \begin_inset Formula $GaN,$ \end_inset \begin_inset Formula $GaP$ \end_inset . The QE is generally higher than that of other photo-cathodes ( \begin_inset Formula $>20$ \end_inset % at UV) but the response time is longer (several hundred ps to ns) limiting their use for ultra short electron bunch generation. Also high currents cannot be drawn without damaging the surface, limiting the peak currents to few amps. They are also very sensitive to chemical contamination and ion back-bombardment must be avoided requiring a vacuum of about \begin_inset Formula $10^{-12}$ \end_inset mbar range. Nevertheless these photo-cathodes are mandatory for polarized electron production, and \begin_inset Formula $GaAs$ \end_inset crystals with different doping are the most popular material for producing polarized electrons when they are illuminated with circularly polarized light \emph on . \layout Section Photo-injector Lasers \begin_inset LatexCommand \label{sec:Photo-injector-Lasers} \end_inset \layout Standard \paragraph_spacing double The electron beam properties depend strongly on the drive laser performance. The length of the laser pulse influences the electron bunch, the energy of the pulse affects the maximum charge of the bunch and the transverse shape of the pulse determines the shape of the electron beam distribution. The typical set up is so called Master Oscillator Power Amplifier(MOPA) \begin_inset Foot collapsed false \layout Standard The term master oscillator power amplifier refers to a configuration of a master laser and an amplifier to boost the output power \end_inset . The oscillator is generally synchronised with a sub-harmonic of the acceleratin g RF voltage with a typical jitter of \begin_inset Formula $\leq1$ \end_inset ps rms. This is followed by an electro-optic selector which produces the temporal structure of the macro bunch or selects a single pulse that is then injected into the amplifier (single, multi pass, regenerative). \layout Standard \paragraph_spacing double Only solid state lasers are able to fulfill the required temporal stability, but they provide laser emission in the infra red range. This emission is converted by frequency multiplication and or mixing in non linear crystals, to produce UV light, which is needed to for photo-emission from metallic and alkali photo-cathodes. The key aspects for the production of a laser beam are: pumping, the laser gain medium, frequency conversion, and the laser beam shape. Let us look at some detail at each of these aspects. \layout Subsection Laser Gain Medium \layout Standard \paragraph_spacing double In context of laser physics, a gain medium is a medium which can amplify the power of a light beam. For the purpose of photo-injector applications, solid state lasers are selected as they have better thermal properties (maximum achievable output power) compared to other gain media like semiconductor or dye lasers. Solid state lasers are based on the solid state gain media such as crystals or glasses doped with rare earth or transition metals,such as Nd:Yag (neodymium doped yttrium aluminium garnet), Yb:Yag (ytterbium doped YAG) or Ti-sapphire (Titanium -Sapphire). \layout Standard \paragraph_spacing double Considerable progress has been made, both for crystals and doping, to provide for lasers which meet the demands of photo-injectors. Ytterbium now tends to replace neodymium as doping ions. \begin_inset Formula $Yb^{3+}$ \end_inset has the advantages of longer lifetime of the upper state giving better energy storage and a low quantum defect ( \begin_inset Formula $1-\frac{\lambda_{laser}}{\lambda_{pump}}$ \end_inset ) which reduces heating during lasing. It allows short pulses and higher power. For passively mode locked applications the high power and short pulses, fiber lasers are now being used to replace crystals. \layout Subsection \paragraph_spacing double Optical Pumping \layout Standard \paragraph_spacing double As the gain medium adds energy to the amplified light, it must receive some energy through a process called pumping which typically involves electrical currents or light inputs i.e. optical pumping. The optical pumping can be done with lasers flash lamps or laser diodes. For photo-injector applications, laser pumping is used to pump Ti:Sapphire with the second harmonic of a Nd:Yag laser (532 nm). Laser diodes have now replaced flash lamps as they provide better efficiency (10% for laser diodes as compared to 1% for flash lamps) and the lifetime is also much longer. On the other hand the emission wavelength of a laser diode is strongly dependent on temperature, which must be accurately controlled. The two pumping families that are of most use are : AlGaAs diodes which emit in the 800 nm region and are used to pump crystals doped with neodymium ions (most popular) and InGaAs diodes emitting around 980 nm and used to pump crystals ytterbium ions which is a promising family. \layout Subsection \paragraph_spacing double Frequency Conversion \layout Standard \paragraph_spacing double This is produced by a non-linear process (multiplication and mixing ) inside the crystals like KDP, KTP, BBo and LiO etc. An efficiency of 50-55% can be expected to covert infrared into green (2nd harmonic generation) and only 25-30% to convert green into UV ( \begin_inset Formula $4^{th}$ \end_inset harmonic generation), giving a total efficiency of about 12-15%. Frequency conversion remains one of the bottlenecks in the progress of photo-injectors. \layout Subsection \paragraph_spacing double Beam Shaping \layout Standard \paragraph_spacing double Beam shaping is the process of redistribution of irradiance through an optical system. Shaping the incident laser beam gives us control over the electron beam shape as the e-beam distribution mimics the laser beam distribution. By irradiating the cathode with an "ideal" incident laser pulse, emittance growth can be optimized. The question of what is the 'ideal' incident beam that would minimise transvers e emittance is still an open one, though some experiments have indicated that the 'top-hat' profile could minimise the transverse emittance under certain conditions. The concept of beam shaping is one of the central themes of this dissertation and is discussed in much more detail in the following chapters. \layout Section \paragraph_spacing double Guns \begin_inset LatexCommand \label{sec:Guns} \end_inset \layout Standard \paragraph_spacing double The electrons produced laser illuminated photo-cathodes have to be accelerated to relativistic energies as quickly as possible in order to minimise the effect of space charge forces. To do this high electric fields are required and a radio-frequency (RF ) is usually the best candidate. Nevertheless, in some cases DC guns or a combination of DC/RF guns is more suitable. \layout Subsection \paragraph_spacing double DC Gun \layout Standard \paragraph_spacing double Applications requiring relatively low electric field (few MV/m) can utilise DC guns, for example the production of polarized electrons from GaAs photo-cath odes. DC guns are limited by their perveance to a production of electron pulses with a peak intensity of a few amps. \layout Subsection \paragraph_spacing double RF Guns \layout Standard \paragraph_spacing double RF guns are now intensively used in many applications from 470 MHz to 17 GHz. In this type of gun the photo-cathode is located on the back wall of an RF cavity. High power microwave drives the cavity to a high electric field. Electric fields higher than 100MV/m can be currently obtained. One of the most popular configuration is the 1.5 cell BNL gun which was upgraded to 1.6 cell for emittance compensation. For producing cw high current up to 100 mA, superconducting RF guns are now being designed. \layout Section \paragraph_spacing double Emittance and Brightness \begin_inset LatexCommand \label{Emittance} \end_inset \layout Standard \paragraph_spacing double There is an intrinsic thermal velocity spread in an electron beam arising from the fact that each point on the source emits electrons with different initial magnitude and direction of the velocity vector. This velocity spread increases downstream of the source as the beam quality deteriorates due to various factors like temperature fluctuations in a plasma source, non linear forces due to external or space charge fields and misalignments of focusing and accelerating elements. To keep track of the quality of the beam, we introduce a figure of merit called \emph on emittance. \layout Standard \paragraph_spacing double To define emittance we observe the dynamics of the particle in phase space. The motion of each particle is defined by three space coordinates \begin_inset Formula $(x,y,z)$ \end_inset and three momentum coordinates ( \begin_inset Formula $P_{x},P_{y},P_{z}$ \end_inset ) at any given instant of time. Let us consider a particle in the x-z plane with total momentum \begin_inset Formula $P$ \end_inset ( \begin_inset Formula $P^{2}=P_{x}^{2}+P_{z}^{2}$ \end_inset ), where \begin_inset Formula $P_{x}\ll P_{z}$ \end_inset and \begin_inset Formula $P_{z}\approx P$ \end_inset . The slope of the trajectory is given by \begin_inset Formula $dx/dz=\dot{x}/\dot{z}\approx P_{x}/P$ \end_inset . At any given distance z along the direction of the beam propagation, every particle is a point in \begin_inset Formula $x-x'$ \end_inset space. This space is called trace space \begin_inset LatexCommand \label{trace_space} \end_inset . One of the simplest definitions of emittance is the area occupied by the particles of the beam in the trace space or \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} A_{x}=\int\int dx'dx\label{eq:Ch3_1}\end{equation} \end_inset \layout Standard \paragraph_spacing double This definition though simple is not useful to indicate beam quality for beams which undergo distortion in shape keeping the trace space area constant. For a more accurate and useful definition of emittance we define the beam as a smooth probability function \begin_inset Formula $f(x,y,x',y')$ \end_inset in x-x' trace space. Here we assume that the particles form a non -interacting ensemble and that the microscopic and macroscopic electromagnetic self fields created by the particles may be neglected for now. \layout Standard The the second moment in the particle coordinates x is given by \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \langle x^{2}\rangle=\int\int\int x^{2}f(x,x',y,y')dx'dxdydy'\label{eq:Ch3_2}\end{equation} \end_inset \layout Standard \paragraph_spacing double and the rms beam width in the x direction is given by \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} x_{rms}=\left(\langle x^{2}\rangle\right)^{1/2}\label{eq:Ch3_3}\end{equation} \end_inset \layout Standard \paragraph_spacing double the rms beam divergence is given by \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} x'_{rms}=\left(\langle x'^{2}\rangle\right)^{1/2}\label{eq:Ch3_4}\end{equation} \end_inset \layout Standard \paragraph_spacing double In similar fashion \begin_inset Formula $\langle xx'\rangle,\langle y^{2}\rangle,\langle yy'\rangle$ \end_inset can also be defined. Then beam emittance is defined as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \tilde{\epsilon}=\left(\langle x^{2}\rangle\langle x'^{2}\rangle-\langle xx'\rangle^{2}\right)^{1/2}\label{eq:Ch3_5eqnForEmittance}\end{equation} \end_inset \layout Standard \paragraph_spacing double The unit of emittance is m-rad. However, since the typical widths and divergence angles of beams are in range of mm and mili-radians, respectively, emittance is usually expressed in units of mm-mrad or cm-mrad. Emittance as described above is an incomplete description of the beam quality. Emittance depends upon on the kinetic energy of the particles and the slope \begin_inset Formula $x'$ \end_inset (and hence the area in trace space \begin_inset Formula $x-x'$ \end_inset ) decreases as longitudinal momentum \begin_inset Formula $P_{z}$ \end_inset increases. Hence one must normalize emittance when comparing beams of different energies. \layout Subsection \paragraph_spacing double Normalized Emittance \begin_inset LatexCommand \label{sub:Normalized-Emittance} \end_inset \layout Standard \paragraph_spacing double For the phase space defined by mechanical momentum \begin_inset Formula $P_{i}$ \end_inset and spatial momentum \begin_inset Formula $q_{i}$ \end_inset , conservation \begin_inset Foot collapsed false \layout Standard Liuovilles' theorem holds. \end_inset of phase space volume can be stated as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \int\int d^{3}q_{i}d^{3}P_{i}=const\label{eq:Ch3_9}\end{equation} \end_inset \layout Standard \paragraph_spacing double While the volume in 6N dimensional phase space (N is number of particles in a beam) is a constant, there is no restriction on the shape to remain the same through time. In fact the particles move through many non-linearities in the fields which causes severe distortions in the shape of the phase space. This often leads to beam blow up and particle loss to nearby walls. The trace space area \begin_inset Formula $A_{x}$ \end_inset is related to the projection of phase space volume into the \begin_inset Formula $x-x'$ \end_inset or equivalently \begin_inset Formula $x-P_{x}'$ \end_inset plane by \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} A_{x}=\frac{1}{P}\int\int dxdP_{x}=\frac{1}{\gamma\beta mc}\int\int dxdP_{x}\label{eq:Ch3_10}\end{equation} \end_inset \layout Standard \paragraph_spacing double In absence of coupling between x-motion and y-motion this area remains constant provided there is no acceleration and deceleration as well, i.e. \begin_inset Formula $(\beta\gamma=const)$ \end_inset . If, however, there is a change of energy ( \begin_inset Formula $\beta\gamma=const)$ \end_inset \begin_inset Formula $A_{x}$ \end_inset and hence \begin_inset Formula $\epsilon_{x}$ \end_inset are no longer constant. Hence one introduces the concept of normalised rms emittance \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \tilde{\epsilon_{n}}=\beta\gamma\tilde{\epsilon}\label{eq:Ch3_11}\end{equation} \end_inset \layout Standard \paragraph_spacing double Normalized emittance is usually a more useful quantity in terms of particle accelerator systems as for an ideal accelerator (linear forces) it remains constant. Growth of normalised emittance is thus an indicator of the presence of beam degrading non -linear forces. \layout Section Transport of Beam \begin_inset LatexCommand \label{sec:Transport-of-beam} \end_inset \layout Standard If we write the general form of the moments integral \begin_inset Formula \begin{equation} \sigma_{ij}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x_{i}x_{j}f_{x}(\vec{x})d^{2}\vec{x}\label{eq:Ch3_2MomentsInetgral}\end{equation} \end_inset \layout Standard where \begin_inset Formula $\vec{x}\equiv(x,x')$ \end_inset and \begin_inset Formula $d^{2}\vec{x}=dxdx'$ \end_inset . Beam evolution through any beam line transport section may be written as \begin_inset LatexCommand \cite{key-74} \end_inset \begin_inset Formula \begin{equation} \sigma_{final}=M\cdot\sigma_{initial}\cdot M^{T}\label{eq:Ch3_10TransportMatrix}\end{equation} \end_inset \layout Standard Based on second moments a set of parameters called the Twiss parameters may now be defined as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \beta_{x}\equiv\frac{\langle x\rangle^{2}}{\epsilon_{x}}\label{eq:Ch3_6}\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \alpha_{x}\equiv\frac{-\langle xx'\rangle}{\epsilon_{x,rms}}\label{eq:Ch3_7}\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \gamma_{x}\equiv\frac{\langle x'^{2}\rangle}{\epsilon_{x,rms}}\equiv\frac{1+\alpha_{x}^{2}}{\beta_{x}}\label{eq:Ch3_8}\end{equation} \end_inset \layout Standard we see that \begin_inset Formula $\sigma$ \end_inset may be expressed as \begin_inset Formula \begin{equation} \sigma=\epsilon_{x,rms}\left[\begin{array}{cc} \beta_{x} & -\alpha_{x}\\ -\alpha_{x} & \gamma_{x}\end{array}\right]\label{eq:Ch3_39SigmaMatrix}\end{equation} \end_inset \layout Standard These parameters provide a means to follow the evolution of the beam. We introduce them here to understand the technique of emittance measurement. \layout Subsection \paragraph_spacing double Measurement of Emittance \begin_inset LatexCommand \label{sub:Measurement-of-Emittance} \end_inset \layout Standard \paragraph_spacing double The input beam emittance sets a limit on the quality of the output beam from an accelerator. Hence it is imperative to have diagnostic devices which can give us a reliable measurement for emittance. The transverse emittance can be measured using lattice optics of a transport line and profile monitor. In general the measurement comprises of determining the Twiss parameters \begin_inset Formula $\epsilon$ \end_inset , \begin_inset Formula $\beta$ \end_inset , \begin_inset Formula $\gamma$ \end_inset and \begin_inset Formula $\alpha$ \end_inset of an incoming beam at position 1, using a monitor at position 2. This can be done by an arrangement comprising of quadrupole [ \begin_inset LatexCommand \ref{cha:AppendixAQuadrupole-as-Focusing} \end_inset ] followed by drift space followed by a monitor. \layout Standard To understand the measurement let us start by considering the evolution of the Twiss parameters by differentiating equations in \begin_inset LatexCommand \ref{eq:Ch3_6} \end_inset , \begin_inset LatexCommand \ref{eq:Ch3_7} \end_inset , and \begin_inset LatexCommand \ref{eq:Ch3_8} \end_inset a drift length after a thin focusing lens (of focal length f) \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \beta'_{x}=2\frac{}{\epsilon_{x}}=-2\alpha_{x}\label{eq:Ch3_12}\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \alpha'_{x}=-\frac{+^{2}}{\epsilon_{x}}=-\gamma_{x}\label{eq:Ch3_13}\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \gamma'_{x}=\frac{2}{\epsilon_{x}}=0\label{eq:Ch3_14}\end{equation} \end_inset \layout Standard \paragraph_spacing double Integrating through the lens of focal length f \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \beta_{xfinal}=\beta_{xo}\label{eq:Ch3_15}\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \alpha_{xfinal}=\alpha_{xo}+\frac{\beta_{xo}}{f}\label{eq:Ch3_16}\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \gamma_{xfinal}=\frac{1+\left(\alpha+\frac{\beta_{xo}}{f}\right)^{2}}{\beta_{xo}}\label{eq:Ch3_17}\end{equation} \end_inset \layout Standard \paragraph_spacing double Now in the drift space we can write \begin_inset Formula \begin{equation} \beta_{x}(z)=\beta_{xo}-2\left(\alpha_{xo}+\frac{\beta_{xo}}{f}\right)(z-z_{o})+\left[\frac{1+\left(\alpha_{xo}+\frac{\beta_{xo}}{f}\right)^{2}}{\beta_{xo}}\right](z-z_{o})^{2}\label{eq:Ch3_18}\end{equation} \end_inset \layout Standard \paragraph_spacing double Multiplying equation \begin_inset LatexCommand \ref{eq:Ch3_18} \end_inset with emittance and rearranging the terms of 1/f, we get an equation for the square of the beam size as function of the focusing strength of the lens. \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \sigma_{x}^{2}(z)=\left[\sigma_{xo}^{2}-2\alpha_{xo}\epsilon_{x}(z-z_{o})+\gamma_{xo}(z-z_{o})^{2}\right]+\frac{2\sigma_{xo}^{2}}{f}\left[\frac{\alpha_{xo}}{\beta_{xo}}(z-z_{o})^{2}-(z-z_{o})\right]+\frac{\sigma_{xo}^{2}}{f^{2}}(z-z_{o})^{2}\label{eq:Ch3_19}\end{equation} \end_inset \layout Standard \paragraph_spacing double Calling the coefficient of \begin_inset Formula $(\frac{1}{f})^{i}$ \end_inset as \begin_inset Formula $m_{i}$ \end_inset (where i=0,1,2), we can write the above equation as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \sigma_{x}^{2}(z)=m_{o}+m_{1}\left(\frac{1}{f}\right)+m_{2}\left(\frac{1}{f}\right)^{2}\label{eq:Ch3_20}\end{equation} \end_inset \layout Standard \paragraph_spacing double The emittance can be obtained by measuring the beam size at a given drift length after a quadrapole magnet, scanning through a range of focusing strengths and extracting \begin_inset Formula $m_{o}$ \end_inset , \begin_inset Formula $m_{1}$ \end_inset and \begin_inset Formula $m_{2}$ \end_inset by fitting parameters to experimental curves. \layout Section \paragraph_spacing double Brightness \begin_inset LatexCommand \label{sec:Brightness} \end_inset \layout Standard \paragraph_spacing double Emittance alone is not enough to define the quality of an electron beam as what truly counts is the total beam current given a particular emittance. The figure of merit which captures that quality is known as Brightness and is commonly defined as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} B=\frac{J}{d\Omega}=\frac{dI}{dSd\Omega}\end{equation} \end_inset \layout Standard \paragraph_spacing double which is the current density per unit solid angle. In this definition brightness varies across the beam. The average brightness can be written as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} B=\frac{cNe}{2\pi l\epsilon_{n,x}\epsilon_{n,y}}\label{Brightness}\end{equation} \end_inset \layout Standard \paragraph_spacing double where N is the number of electrons; l is the rms bunch length, and \begin_inset Formula $\epsilon_{n,x}$ \end_inset and \begin_inset Formula $\epsilon_{n,y}$ \end_inset are normalised rms emittance of the electron beam. This is the customary definition of brightness with units in \begin_inset Formula $\frac{Amp}{(mm-mrad)^{2}}$ \end_inset . Though customary, it is not a universal definition and brightness may alternati vely be defined as \begin_inset Formula \begin{equation} B=\frac{N_{e}}{\epsilon_{n,x}\epsilon_{n,y}}\label{eq:Ch3_26}\end{equation} \end_inset where the units of brightness are \begin_inset Formula $m^{-3}$ \end_inset . \layout Section \paragraph_spacing double FEL-Applications \begin_inset LatexCommand \label{sec:FEL-Applications} \end_inset \layout Standard \paragraph_spacing double The universal gain parameter \begin_inset Formula $\rho$ \end_inset for a free electron laser is a simple function of the beam brightness \begin_inset LatexCommand \cite{key-23} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \[ \rho\sim B^{1/3}.\] \end_inset \layout Standard \paragraph_spacing double The appeal of photo-injectors for FELS is that a very high brightness beam can be produced. RF photo-injectors are most suitable for this application as they can provide high brightness beams directly from the gun compared to alternative high brightness injector systems such a pulsed girded thermionic gun followed by RF longitudinal bunching system and a damping ring to reduce transverse emittance. Also high current densities at the cathode are desirable in order to minimise transverse emittance. Thermionic cathodes are limited to \begin_inset Formula $10$ \end_inset \begin_inset Formula $A/cm^{2}$ \end_inset whereas photo cathodes can easily produce peak current densities that much higher: more than 3000 \begin_inset Formula $A/cm^{2}$ \end_inset has been extracted at CTF, CERN, by illuminating a \begin_inset Formula $Cs_{2}Te$ \end_inset cathode with a Nd:YLF laser system. The high electric fields produced by the RF guns are necessary to both extract the high currents and to accelerate the beam to relativistic energies where the effect of the space charge forces, scaling as \begin_inset Formula $1/\gamma^{2}$ \end_inset , become negligible. \layout Standard \paragraph_spacing double Proposals for increasingly shorter wavelength linac-based Free Electron Lasers such as the Linac Coherent Light Source (LCLS) at SLAC \begin_inset Foot collapsed false \layout Standard Stanford Linear Accelerator Center \end_inset and the VUV FEL at the TESLA test facility at DESY need electron sources with much higher brightness than are currently available. Consequently there is great motivation to better understand physics of photo-injectors and to improve all related technologies. \layout Chapter \paragraph_spacing double Problem and the Proposed Solution \layout Section Introduction \layout Standard \paragraph_spacing double The quality of an electron beam is characterised, for most practical purposes, by its brightness B: \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} B=\frac{N_{e}}{2\pi l\epsilon_{n,x}\epsilon_{n,y}}(\ref{Brightness})\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $N_{e}$ \end_inset is the number of electrons, \begin_inset Formula $l$ \end_inset is the rms electron bunch length, \begin_inset Formula $\epsilon_{n,x}$ \end_inset and \begin_inset Formula $\epsilon_{n,y}$ \end_inset are normalized rms emittance of the electron beam. It is evident from equation 4.1 that emittance growth adversely affects brightness. Thus to achieve high brightness of an electron beam, we need to fully comprehen d the reasons behind transverse emittance growth and to suppress this growth as much as possible. In section \begin_inset LatexCommand \ref{sec:4.2_Emittance-Growth-of} \end_inset we take a look at various contributions to emittance growth. In section \begin_inset LatexCommand \ref{sec:4.3_Space-Charge-Effect} \end_inset we discuss the effect of space charge forces on emittance. This emphasis on space charge dominated emittance is warranted as beam emittance in photo-injectors is primarily dominated by linear and non-linear space charge effects. The linear space charge forces can be reasonably well compensated for by solenoid focusing, but non linear space charge effects continue to impact the transverse emittance of the beam. Non-linear space charge effects primarily arise from the non-uniformity of the electron distribution. Section \begin_inset LatexCommand \ref{sec:4.4_Emittance-Growth} \end_inset deals with the emittance growth due to the non uniformity of space charge distribution and attempt to comprehend the underlying reasons of the non uniformity. It turns out there are two major sources of non-uniformities in photo-cathode guns: (a) non uniform quantum efficiency (QE) on the photo-cathode surface and (b) the non uniformity due to laser pulse that illuminates the cathode surface. Finally in section \begin_inset LatexCommand \ref{cap:Ch4-beamShaping} \end_inset we take a look at the concept of beam shaping as an effective method to compensate for the non-uniformities in the electron beam distribution. Shaping the laser beam that illuminates the photo-cathode has the potential to correct for both kinds of non-uniformities. \layout Section \paragraph_spacing double Emittance Growth of Electron Beams in Photo-Injectors \begin_inset LatexCommand \label{sec:4.2_Emittance-Growth-of} \end_inset \layout Standard \paragraph_spacing double The emittance of the electron beam in photo-injectors is affected by three contributions:thermal emittance,radio frequency (RF) induced emittance, and space charge force induced emittance. The total emittance of the photo-injector can be approximated as \begin_inset Foot collapsed false \layout Standard This expression is an approximation as this neglects correlations betweens terms that may exist \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \epsilon_{n}=\left[\left(\epsilon_{n}^{th}\right)^{2}+\left(\epsilon_{n}^{rf}\right)^{2}+\left(\epsilon_{n}^{sp}\right)^{2}\right]^{1/2}\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $\epsilon_{nx}^{th}$ \end_inset is the thermal emittance, \begin_inset Formula $\epsilon_{nx}^{rf}$ \end_inset is the transverse emittance arising from the fact that electrons at different longitudinal positions near the exit receive different transverse kicks due to variation of the RF field and \begin_inset Formula $\epsilon_{nx}^{sp}$ \end_inset is the emittance attributed to the repulsive force due to space charge effect. The thermal emittance,the lower limit of the emittance in a photo-cathode gun ( \begin_inset LatexCommand \cite{key-67} \end_inset ) can be written as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \epsilon_{n,th}=r_{c}\left(\frac{kT_{e}}{m_{o}c^{2}}\right)^{1/2}\end{equation} \end_inset \layout Standard \paragraph_spacing double For a laser driven photo-cathode, \begin_inset Formula $r_{c}$ \end_inset is the rms radius of the laser spot on the cathode and \begin_inset Formula $T_{e}$ \end_inset is the effective temperature \begin_inset Foot collapsed false \layout Standard The effective temperature can be derived from Schottky effect measurements and is different from simply the temperature of the cathode. For high QE cathodes, a value of 0.2 eV is found. \end_inset of the photoelectrons. This thermal emittance is the lower limit of emittance and in practise the emittance is usually higher than this as the RF and the space charge fields contribute significantly toward increasing the emittance. The RF induced emittance growth for a bunch of negligible charge is given by \begin_inset LatexCommand \cite{key-23} \end_inset \begin_inset Formula \begin{equation} \epsilon_{n,i}^{rf}=\frac{eE_{o}}{2\sqrt{2}m_{o}c^{2}}k^{2}\sigma_{i}^{2}\sigma_{z}^{2}\end{equation} \end_inset where \begin_inset Formula $k=2\pi/\lambda$ \end_inset is the wave number of the RF field, \begin_inset Formula $E_{o}$ \end_inset is the maximum RF field at the cathode, \begin_inset Formula $\sigma_{z}$ \end_inset is the longitudinal rms bunch length and \begin_inset Formula $\sigma_{i}$ \end_inset is the transverse rms size where i can be x or y. Clearly the RF contribution increases by increasing initial acceleration. \layout Section \paragraph_spacing double Space Charge Effect on Emittance \begin_inset LatexCommand \label{sec:4.3_Space-Charge-Effect} \end_inset \layout Standard \paragraph_spacing double Particles in an intense beam are under the influence of strong electromagnetic forces. The mutual interaction of charged particles in a beam can be considered as a sum of "collisional" and "smooth" forces. The "collisional" part of the force primarily arises if the particle feels the effect of its immediate neighbours dependent of their individual positions. This causes random displacement of particle trajectory and statistical fluctuations in the beam distribution. On the contrary if we can ignore any statistical effects (arising due to the fact that the beam is composed of many particles) and approximate the motion of a single test particle as being under the influence of a smooth force due surrounding "space charge", such an effect is called space charge effect. Under this condition, assuming that the spread in electron velocities is small, the space charge potential obeys Poisson equation the interaction between the particles can be described in terms of a smoothed out force. This force maybe treated in the same way as external forces acting on the beam. For non neutral beams the terms space charge and self fields maybe used interchangeably since the moving space charge of the particle distribution is the source for both the electric and magnetic self fields. \layout Standard \paragraph_spacing double A fundamental parameter in plasma physics called Debye length, \begin_inset Formula $\lambda_{D}$ \end_inset , and maybe used as measure of the relative importance between the "collisional" and the "smooth" force description of space charges. In case of neutral plasma with temperature \begin_inset Formula $T$ \end_inset and equal positive and negative ion densities \begin_inset Formula $n$ \end_inset , the excess potential set up by a test charge is screened off at a distance \begin_inset Formula $\lambda_{D}$ \end_inset by the charge redistribution in the plasma. For a non-relativistic plasma, Debye length \begin_inset Formula $\lambda_{D}$ \end_inset is defined,by the ratio of the rms velocity \begin_inset Formula $\tilde{v_{x}}=\overline{\left(v_{x}^{2}\right)}^{2}$ \end_inset and the plasma frequency \begin_inset Foot collapsed false \layout Standard plasma frequency \begin_inset Formula $\omega_{p}^{2}=\frac{ne^{2}}{\epsilon_{o}m}$ \end_inset is a fundamental time scale in plasma physics. \begin_inset Formula $\omega_{p}$ \end_inset corresponds to the typical electrostatic oscillation frequency of a given species in response to a small charge separation. \end_inset \begin_inset Formula $\omega_{p}$ \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \lambda_{D}=\frac{\tilde{v_{x}}}{\omega_{p}}\end{equation} \end_inset \layout Standard \paragraph_spacing double A charged particle beam maybe considered as a non neutral plasma and the analogy extended to the fact that a local perturbation in the equilibrium distribution of beam with temperature \begin_inset Foot collapsed false \layout Standard The transverse temperature of a charged particle beam for a thermal distribution is \begin_inset Formula $\gamma m\left(\tilde{v_{x}}\right)^{2}=k_{B}T$ \end_inset where \begin_inset Formula $\tilde{v_{x}}$ \end_inset is rms velocity and \begin_inset Formula $k_{B}$ \end_inset is the Boltzmann constant. \end_inset T and density n, confined by external fields will be screened off again at a distance \begin_inset Formula $\lambda_{D}$ \end_inset . But for a relativistic charged particle beam the transverse motion is expressed as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \ddot{x}=\omega_{p}^{2}x=\frac{F_{s}}{\gamma m}\end{equation} \end_inset \layout Standard \paragraph_spacing double Since the electric coloumbic repulsion is reduced by magnetic attraction \begin_inset Formula \begin{equation} F_{s}=eE\left(1-\beta^{2}\right)=\frac{eE_{s}}{\gamma^{2}}\end{equation} \end_inset \layout Standard \paragraph_spacing double If we assume uniform density \begin_inset Formula $E_{s}\sim enx$ \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{eqnarray} \omega_{p}^{2} & = & \frac{e^{2}n}{\epsilon_{o}\gamma^{3}m}\\ \omega_{p} & = & \left(\frac{e^{2}n}{\epsilon_{o}\gamma^{3}m}\right)^{1/2}\end{eqnarray} \end_inset \layout Standard \paragraph_spacing double Using equation 4.9 for the plasma frequency and assuming \begin_inset Formula $\tilde{v_{x}}\ll c$ \end_inset , i.e. the transverse motion is non relativistic, the Debye length for a relativistic beam maybe defined as \begin_inset Formula \begin{equation} \lambda_{D}=\frac{\tilde{v_{x}}}{\omega_{p}}=\left(\frac{\epsilon_{o}\gamma^{3}\tilde{(v_{x}})^{2}}{e^{2}n}\right)^{1/2}\end{equation} \end_inset \layout Standard \paragraph_spacing double Using the relation \begin_inset Formula $\gamma m\left(\tilde{v_{x}}\right)^{2}=K_{B}T$ \end_inset to substitute \begin_inset Formula $\tilde{v_{x}}=\left(\frac{K_{B}T}{\gamma m}\right)^{1/2}$ \end_inset in the above equation to get \begin_inset Formula \begin{equation} \lambda_{D}=\left(\frac{\epsilon_{o}\gamma^{2}K_{B}T}{e^{2}n}\right)\label{eq:Ch4_DebyeLength}\end{equation} \end_inset \layout Standard \paragraph_spacing double If the Debye length is large compared to the beam radius \begin_inset Formula $\left(\lambda_{D}\gg a\right)$ \end_inset the screening is ineffective and single particle behaviour dominates. On the contrary is the Debye length is small compared to beam radius \begin_inset Formula $\left(\lambda_{D}\ll a\right)$ \end_inset the collective effects due to space charge forces will be significant. From equation \begin_inset LatexCommand \ref{eq:Ch4_DebyeLength} \end_inset it is evident that Debye length increases with increasing energy \begin_inset Formula $(\gamma)$ \end_inset so at sufficiently high energies the space charge forces become insignificant compared to external forces. \layout Standard \paragraph_spacing double In the photo-injectors especially near the cathode surface the beam is not yet at very high energy and is hence dominated by space charge effects. Analytical expressions for space-charge growth have been derived by Kim ( \begin_inset LatexCommand \cite{key-23} \end_inset ). If the aspect ratio \begin_inset Formula $A=\frac{\sigma_{x}}{\sigma_{z}}$ \end_inset of the bunch is less than 1, then the transverse emittance growth due to space charge is given by \begin_inset Formula \begin{equation} \epsilon_{n,i}^{sp}=\frac{AQ}{8\sqrt{2\pi}\epsilon_{o}c\sigma_{i}E_{o}sin\phi_{o}}\mu_{i}(A)\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \mu_{i}(A)=\sqrt{\langle E_{i}^{2}\rangle\langle i^{2}\rangle-\langle E_{i}i\rangle^{2}}\end{equation} \end_inset \layout Standard \paragraph_spacing double Here \begin_inset Formula $Q$ \end_inset is the total charge in the bunch, \begin_inset Formula $E_{o}$ \end_inset is the maximum RF field at the cathode, \begin_inset Formula $\phi_{o}$ \end_inset is the RF phase at the extraction and \begin_inset Formula $\mu_{i}(A)$ \end_inset is known as the transverse space-charge factor where i is either x or y. This calculation has an inherent assumption that the beam is in the linear space charge regime, i.e. the space charge forces arising from uniform space charge distribution ( \begin_inset Formula $\rho_{o}=const$ \end_inset ) vary linearly from the center of the beam. In the linear regime the emittance growth that the electron beam experiences can be compensated by emittance compensation techniques ( \begin_inset LatexCommand \ref{sub:Emittance-Compensation} \end_inset ). \layout Section \paragraph_spacing double Emittance Growth \begin_inset LatexCommand \label{sec:4.4_Emittance-Growth} \end_inset \layout Standard \paragraph_spacing double Charged particle beams in equilibrium states, i.e. states in which the particle distribution remains stationary with distance, can be represented by Maxwell Boltzmann distribution. This distribution, also called thermal distribution, is defined by \begin_inset Formula $f(\overrightarrow{x,}\overrightarrow{P})=f_{o}exp(-H/K_{B}T)$ \end_inset , where H is single particle Hamiltonian. A continuous beam can thus be treated as a transverse Maxwell-Boltzmann distribution for which the particle density obeys the following Boltzmann relation (ref 2.2.1) \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} n(r)=n_{o}exp[-q\phi(r)/K_{B}T_{\perp}]\end{equation} \end_inset \layout Standard \paragraph_spacing double Here \begin_inset Formula $n_{o}$ \end_inset is the density at \begin_inset Formula $r=0$ \end_inset , \begin_inset Formula $T_{\perp}$ \end_inset is the transverse laboratory temperature of the beam, \begin_inset Formula $K_{B}$ \end_inset is the temperature constant and \begin_inset Formula $\phi(r)$ \end_inset is sum of the effective external potential \begin_inset Formula $\phi_{e}(r)$ \end_inset and effective potential due to self fields \begin_inset Formula $\phi_{s}(r)(1-\beta^{2})$ \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \phi=\phi_{e}(r)+\phi_{s}(r)(1-\beta^{2})\end{equation} \end_inset \layout Standard \paragraph_spacing double In the space charge dominated regime (low \begin_inset Formula $\gamma)$ \end_inset , the repulsive space charge forces become comparable in strength to external forces i.e. \begin_inset Formula $\left[q\phi_{s}\left(1-\beta^{2}\right)=-q\phi_{e}\right]$ \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{eqnarray} n(r) & = & n_{o}=const\textnormal{ for $r\le$a}\\ n(r) & = & 0\textnormal{ for $r>a$}\end{eqnarray} \end_inset \layout Standard \paragraph_spacing double Such an equilibrated, uniform beam can be considered stationary and does not suffer emittance growth due to non linear space charge forces. \layout Standard \paragraph_spacing double Real laboratory beams are usually not in perfect equilibrium state and there are a large number of effects that can cause emittance growth. For example non linearities in the applied forces, collisional forces, beam mismatch causing oscillations of the rms radius and non linear space charge forces arising from non uniform beam density profiles. Of these the study and possible rectification of emittance growth due to non linear space charge forces arising from non uniformity in beam profiles of primary interest to this thesis. \layout Standard \paragraph_spacing double As of now analytical theory is not available to model non linear space charge forces arising out of nonuniform electron beam distribution. The only available tools to study the phenomenon are thus experiments and numerical simulations. Unlike for emittance growth due to linear space charge forces, compensation techniques are not yet available hence we begin by looking at the underlying reasons of non uniformity. There are two primary sources of non uniformity in electron beam distribution as discussed below. \layout Subsection \paragraph_spacing double Quantum Efficiency Variation on the Photo-cathode \begin_inset LatexCommand \label{QE} \end_inset \layout Standard \paragraph_spacing double Quantum efficiency (QE) is the ratio of number of electrons emitted by the photo cathode to the number of incident photons. \begin_inset Formula \[ QE=\frac{electrons\left(emitted\right)}{photons\left(incident\right)}\] \end_inset The transverse charge distribution of the electron beam is a convolution \begin_inset Foot collapsed false \layout Standard A convolution is an operator that expresses the amount of overlap of one function as it is shifted over another function \end_inset of the laser pulse transverse energy density and the two dimensional quantum efficiency map of the photo-cathode. \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \rho_{\perp}=\Omega\otimes\delta\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $\Omega$ \end_inset is the transverse energy density of the laser pulse, \begin_inset Formula $\delta$ \end_inset is the two dimensional QE map, and \begin_inset Formula $\otimes$ \end_inset represents convolution. Any coarse grained nature of the QE map would also make its way into the electron beams transverse distribution. Hence a uniform QE across the surface of the cathode is a requirement toward achieving a uniform electron beam. \layout Standard \paragraph_spacing double But quantum efficiency deteriorates as the metal surface is contaminated by insufficient vacuum and or RF breakdowns. Thus there is a need to study the quantum efficiency distribution over the surface of the cathode and to correct for any such non-uniformity which may occur over the life time of the cathode. \layout Subsection \paragraph_spacing double Non-Uniformity in the Laser Illumination. \begin_inset LatexCommand \label{sub:Non-uniformity-in-the} \end_inset \layout Standard \paragraph_spacing double A nonuniform laser beam generates a non uniform electron beam distribution. A non uniform electron beam distribution experiences emittance growth on a time scale of a plasma period. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/graph_zhou.eps display none scale 50 keepAspectRatio clip \end_inset . \layout Caption Laser mask experiment at ATF, BNL. \layout Standard Distribution of transverse laser beam slices along a beam diameter for the various distribution types. Each distribution type corresponds to a different laser masks. The measured data is in red ,the simulated data in blue and the analysis in black lines.The graph is reproduced from \begin_inset LatexCommand \cite{key-75} \end_inset \begin_inset LatexCommand \label{cap:Ch4_ZhouLaserDist} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset The effect of controlled non uniformity of the laser illumination on emittance was studied at the Accelerator Test Facility (ATF) at Brookhaven National Laboratory (BNL). These studies comprised of both experiments and numerical simulations. Experimentally non uniform electron beam distributions were created by introducing an artificially nonuniform laser beam by using special masks. Using four different laser masks four types of cylindrically symmetric laser beams with different non uniformities are produced. The four different laser masks created a peak to peak intensity distribution of 20%,45%,65% and 75% respectively in the incident laser beam and were respectively named type 2 through type 5. The full beam without any masks was marked as type 1. The intensity distributions of the laser beam with these masks is shown in figure \begin_inset LatexCommand \ref{cap:Ch4_ZhouLaserDist} \end_inset . \layout Standard \paragraph_spacing double The normalized transverse emittance was measured ( \begin_inset LatexCommand \ref{slice-emittance} \end_inset ) for a charge of 0.48 nC using the above mentioned masks on the laser. The experimental results showed that the emittance grows by about 30% for a 45% intensity variation to about a factor of 2 in more extreme distortions of the beam. Numerical simulations agreed well with the experimental results. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/emittance_vs_lasermasks.eps lyxscale 50 display none scale 50 BoundingBox 0bp 100bp 306bp 336bp clip \end_inset \layout Caption Emittance vs. laser distribution types. \layout Standard graphs reproduced from \begin_inset LatexCommand \cite{key-75} \end_inset \begin_inset LatexCommand \label{ZhouEmittancevsLaser} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The experiment concluded that emittance grows as a function of transverse laser distribution non uniformity and for cylindrically symmetric electron beams, there is a a linear dependence of the emittance on the bunch charge for various degrees of non uniformity. This growth in emittance due to the non uniformity in the laser beam distributi on can be analysed in terms of the free energy of the electron beam as discussed in the next section. \layout Section \paragraph_spacing double Free Energy and Emittance Growth \begin_inset LatexCommand \label{sec:4.5_Free-Energy-and} \end_inset \layout Standard \paragraph_spacing double Any deviation from the uniform density profile of a space charge dominated Maxwell-Boltzmann distribution causes emittance growth. This is because a non stationary initial beam has higher total energy per particle compared to a stationary beam. This energy difference \begin_inset Formula $\bigtriangleup E$ \end_inset between a stationary and a non-stationary beam represents free energy that can be thermalised by non-linear space charge forces and thus lead to emittance growth. The energy difference between a non-stationary and a stationary beam is given by \begin_inset LatexCommand \cite{key-76} \end_inset \begin_inset Formula \begin{equation} \frac{\epsilon_{nf}}{\epsilon_{ni}}=\left(1+\frac{Nr_{c}\tilde{x}}{15\sqrt{5}\gamma_{o}\epsilon_{ni}^{2}}\frac{U}{w_{o}}\right)\end{equation} \end_inset where \begin_inset Formula $\epsilon_{ni}$ \end_inset , \begin_inset Formula $\epsilon_{nf}$ \end_inset are the initial and final normalised emittances, respectively, N is the number of particles in one bunch, \begin_inset Formula $r_{c}$ \end_inset is the classical electron radius, \begin_inset Formula $\tilde{x}$ \end_inset is the rms transverse dimension and \begin_inset Formula $U/w_{o}$ \end_inset is the normalized field energy difference per unit length between non-uniform and uniform initial beams. \layout Standard At this point it is important to emphasize that the equation 4.19 defines the \emph on possible \emph default emittance growth due to free energy. This is the excess free energy that maybe thermalized. This predicted growth occurs only in the presence of non linear external or space charge effects. \begin_inset Foot collapsed false \layout Standard In storage rings,the very long lifetime ensures that stochastic effects such as RF noise or Coulomb collision thermalise the beam. \end_inset \layout Section Proposed Solution \layout Subsection Beam Shaping for Emittance Optimization \begin_inset LatexCommand \label{sub:4.6_Beam-Shaping-for} \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/beam_shaping1.eps lyxscale 40 display none clip \end_inset \layout Caption Illustration showing redistribution of Gaussian profile into a tophat profile. \begin_inset LatexCommand \label{cap:Ch4-beamShaping} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double Beam shaping is the redistribution of the irradiance and phase profile of the output of a laser beam. In a laser beam the shape of the beam i.e. the transverse distribution is determined by the irradiance properties and the propagation properties i.e. the longitudinal distribution is determined by the phase profile. Longitudinal distribution of a laser beam, often referred to as pulse shaping, can be achieved by use of gratings, pockel cells and techniques like frequency domain modulation. The transverse laser profile can be controlled in various ways from brute force truncation to more elaborate techniques of deformable mirrors and active laser shaping. For the purpose of this thesis, we will refer to longitudinal beam distribution as pulse shaping and reserve the term beam shaping for spatial/transverse distribution of the laser beam. \layout Standard \paragraph_spacing double A spatial light modulator (SLM) may be used to shape the laser beam irradiating the cathode surface. Shaping of the laser beam can be useful to improve electron beam brightness as the technique may be used to reduce emittance growth. Emittance growth can be reduced in in two ways: by compensating for quantum efficiency variations on photo-cathode surface or by providing the "ideal" shape of the laser distribution to irradiate the photo-cathode. Quantum efficiency variations can be minimized, by irradiating less efficient sections of the cathode with more intense light and vice versa. To find the most suited laser distribution, it is possible to use algorithms parametrised by emittance \begin_inset LatexCommand \cite{key-73} \end_inset . \layout Standard \paragraph_spacing double Digital Light Processing technique using the digital mirror device (DMD) kit may be used as a spatial light modulator. The kit comprising of an array or mirrors and underlying control electronics can be used to shape the transverse profile of the laser beam as well as control the illumination on the photo-cathode. This provides control over the distribution of the electron beam. The control over the electron beam distribution can be used to optimize emittance. \layout Section Summary \layout Standard \paragraph_spacing double Electron beam brightness can be improved if the emittance of the e-beam is not allowed to grow. Emittance growth in electron beams occurs due to presence of non-linear forces. These non-linear forces must be suppressed to check emittance growth. Since it is not possible to compensate for non-linear growth of the emittance, we look at reasons of non-linearity and if possible try correct for these. The non-linearities arise if the electron beam distribution is not uniform. To suppress the non linear forces it is imperatives to start with and maintain a uniform electron beam distribution. Since the uniformity of electron beam distribution is affected by the QE of the photo cathode and the uniformity of the drive laser distribution, we propose to examine both the issues in depth using the technology of digital light processing. \layout Chapter Quantum Efficiency Measurements \layout Section \paragraph_spacing double Introduction \begin_inset LatexCommand \label{sec:IntroductionChapter5} \end_inset \layout Standard \paragraph_spacing double In this chapter we discuss the technology that we use for emittance optimization of electrons beams,namely digital light processing. In section \begin_inset LatexCommand \ref{sec:DLPOverview} \end_inset we give an overview of the digital light processing technology. In section \begin_inset LatexCommand \ref{sec5.3:dmd} \end_inset we explain the working of the key element of the technology namely digital mirror device of DMD. Section \begin_inset LatexCommand \ref{sec:Ch5_SoftwareDMD} \end_inset is a discussion of the device control and the software development that was needed to ensure the smooth functioning of the device for our experiments. Section \begin_inset LatexCommand \ref{sec:Ch5_ExperimentSetup} \end_inset gives a schematic sketch of the experiment and reviews the experimental procedure followed to measure the QE of the ATF photo-cathode. Section \begin_inset LatexCommand \ref{sec:Ch5_New-QETechnique} \end_inset is a discussion of the new technique to measure QE and generate spatial QE maps of the photo-cathode surface using digital light processing. This section also has representative plots and an overview of results obtained by applying this technique at the ATF. In section \begin_inset LatexCommand \ref{sec:Ch5_Laser-Cleaning-Cathode} \end_inset we review the standard technique of cleaning the contaminated surface of a photo-cathode and present the results in subsection \begin_inset LatexCommand \ref{sub:Ch5_Results} \end_inset of the the measurement of QE for both before and after cleaning the photo-catho de at ATF. \layout Section \paragraph_spacing double Overview of Digital Light Processing \begin_inset LatexCommand \label{sec:DLPOverview} \end_inset \layout Standard \paragraph_spacing double Digital Light Processing \begin_inset Formula $(DLP^{TM})$ \end_inset \begin_inset Foot collapsed false \layout Standard DLP is registered trademark of Texas Instruments. \end_inset ,developed by Texas instruments,is a new projection and display technology. The key element in this technology is the digital mirror device or DMD. The DMD is a micro-electromechanical system (MEMS) and combined with image processing, memory, light source and optics forms the DLP system. It is a MEMs consisting of hundreds of thousands of micro-mirrors, moving mechanically, controlled by underlying CMOS [ \begin_inset LatexCommand \ref{Appendix_CMOS} \end_inset ] electronics as shown in figure \begin_inset LatexCommand \ref{cap:Ch5_F1DMD} \end_inset . The mirrors are highly reflective and are used to modulate light thus making DMD optical MEMs \begin_inset LatexCommand \ref{dmd} \end_inset . The DLP system is capable of projecting large, bright and high contrast images and is a competing technology with liquid crystals for the state of the art in the arena of display technologies for movie theatres and high definition television. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/dmd1.eps display none \end_inset \layout Caption Schematic of two DMD mirror pixel next to a typical DMD consisting of 1024x768 individually addressable mirror-pixels. \begin_inset LatexCommand \label{cap:Ch5_F1DMD} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset Apart from display technologies the DMD finds many scientific applications in varied fields of optical signal processing and lithography. The DLP is also widely used in spectroscopy ( \begin_inset LatexCommand \cite{key-25} \end_inset ) and microscopy ( \begin_inset LatexCommand \cite{key-26} \end_inset ) as a spatial light modulator. We also exploit the properties of the DMD as a spatial light modulator to study emittance optimization in electron beams. \layout Section \paragraph_spacing double Digital Mirror Device \begin_inset LatexCommand \label{sec5.3:dmd} \end_inset \layout Standard \paragraph_spacing double Digital Light Processing is a technique to output optical words from an input of digital words. The key element in this technology is the digital mirror device or DMD. The DMD, is micro electro-mechanical system (MEMS), which in standard format, comprises of 1024 x768 individually addressable micro-mirrors. The mirrors are highly reflective and are used to modulate light thus making DMD an optical MEMs and more specifically a reflective spatial light modulator (SLM). These mirrors rest on hinges and are mounted on SRAM \begin_inset Foot collapsed false \layout Standard Static random access memory \end_inset cell. The hinges permit a mirror to rotate a \begin_inset Formula $\pm12^{o}$ \end_inset from its rest position. A digital word i.e. an array of ones and zeros is used as an input. Depending upon whether a one or a zero addresses the SRAM underlying a mirror pixel the mirror turns a \begin_inset Formula $+12^{o}$ \end_inset (ON state) or \begin_inset Formula $-12^{o}$ \end_inset (OFF state). If co-ordinated with a light source, such that a mirror in the ON state reflects light toward a pupil of the projection lens and a mirror in the OFF state directs light away from the projection lens, an optical image of the input digital word can be formed. \layout Standard \paragraph_spacing double In this application DLP \begin_inset Formula $^{TM}$ \end_inset is used as spatial light modulator. High fidelity images, of an input digital word are used to irradiate the cathode. Control over the shape of the digital input image translates into control over the laser beam. \layout Subsection \paragraph_spacing double DMD as a Spatial Light Modulator(SLM) \begin_inset LatexCommand \label{sub:DMD-asSLM} \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/dmd_AS_SLM.eps display none \end_inset \layout Caption Blazed condition of pixelated mirror. \begin_inset LatexCommand \label{cap:Ch5_F2BlazedCondition} \end_inset \layout Standard (a) shows the incident light hitting the mirrors with periodicity d in flat state. In this case most of the light is concentrated in the first order of diffractio n. (b) The mirrors are in +1 state (ON) state.This is the blazed grating condition and most of the diffracted radiation is in the 2nd order ( as shown here), producing a coherent light modulator. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double A pixelated spatial light modulator is compromised of discrete elements and can be constructed as a transmissive or reflective device. The DMD can be used as a pixelated spatial light modulator with each mirror acting as a pixelated reflector. This pixelated reflector behaves like a diffraction grating in a direction \begin_inset Formula $\theta_{r},$ \end_inset relative to surface normal. The grating condition is determined by the pixel period, d, the wavelength \begin_inset Formula $\lambda$ \end_inset , and the angle of incidence, \begin_inset Formula $\theta_{i}$ \end_inset . Figure \begin_inset LatexCommand \ref{cap:Ch5_F2BlazedCondition} \end_inset shows the optical layout in which the maxima in reflectivity distribution is governed by diffraction \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} d(sin\theta_{r}+sin\theta_{i})=n\lambda\label{eq:Ch5_1gratingCond}\end{equation} \end_inset where n is the order of diffraction. The condition in which the angle of incidence and the angle of diffraction are identical is referred to as Littrow configuration and the diffraction equation reduces to the Bragg condition \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} 2dsin\theta=n\lambda\label{eq:Ch5_2}\end{equation} \end_inset \layout Standard \paragraph_spacing double The tilt angle of the mirrors strongly controls the reflective power. The Fraunhoffer diffraction in the Littrow case directs light into a ray with an angle equal to the angle of incidence. \layout Subsection ATF Calculation \begin_inset LatexCommand \label{sub:ATF-Calculation} \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/blaze_calculation.eps display none \end_inset \layout Caption Graph showing the calculation of the angle of incidence for the DMD at the ATF. \begin_inset LatexCommand \label{cap:Ch5_ATF_DMD_Calulate} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double To calculate the correct angle of incidence for the DMD device the following calculation was done. The wavelength of the beam incident upon the DMD is 532 nm, the pitch of the micro-mirrors is 13.6 \begin_inset Formula $\mu m$ \end_inset and the tilt angle is a \begin_inset Formula $\pm12^{o}$ \end_inset . Considering these parameters the grating equation is written down as in \begin_inset LatexCommand \ref{eq:Ch5_1gratingCond} \end_inset . Now plotting \begin_inset Formula $\theta_{d}$ \end_inset as well as \begin_inset Formula $\theta_{i}$ \end_inset as a function of the angle of incidence \begin_inset Formula $\theta_{i}$ \end_inset , we find the angle where \begin_inset Formula $\theta_{d}$ \end_inset and \begin_inset Formula $\theta_{r}$ \end_inset coincide for a particular order. For \begin_inset Formula $\lambda=532.10^{-9}m$ \end_inset , \begin_inset Formula $c=2.99.10^{8}\frac{m}{s}$ \end_inset and \begin_inset Formula $d=13.6.10^{-6}m$ \end_inset , the equations are \begin_inset Formula \begin{equation} \sin(\theta_{i}+\theta_{r})=\frac{n\lambda}{d}\label{eq:Ch5_3}\end{equation} \end_inset \begin_inset Formula \begin{equation} \theta_{r}=-\theta_{i}+2.\theta_{m}\label{eq:Ch5_4}\end{equation} \end_inset \layout Standard As shown in figure \begin_inset LatexCommand \ref{cap:Ch5_ATF_DMD_Calulate} \end_inset , this angle of incidence was found to be \begin_inset Formula $35^{o}$ \end_inset for the 10th order, where equation \begin_inset LatexCommand \ref{eq:Ch5_4} \end_inset is the blazing condition. \layout Section \paragraph_spacing double Software Development for DMD \begin_inset LatexCommand \label{sec:Ch5_SoftwareDMD} \end_inset \layout Standard \paragraph_spacing double A control system was needed for the local and remote control of the digital mirror device. A client server \begin_inset Foot collapsed false \layout Standard Client/Server is a network architecture which separates the client (often a graphical user interface) from the server. A client is software system that accesses a (remote) service on another computer by some kind of network. An application server is a server computer in a network dedicated to running certain applications. In our case the client runs from a control room computer and the server runs on Win term connected by USB to the DMD. \end_inset model windows application using windows sockets was created for this purpose. The server application resides on a thin windows terminal connected to the DMD device via USB. The DMD is controlled via this server. The server calls upon an activeX \begin_inset Foot collapsed false \layout Standard A small computer program that can be plugged into compliant applications to provide added functionality. The controls have an .ocx file name extension. \end_inset to perform basic control operations on the DMD. These primary control operations comprise of ensuring that the device is detected by the controlling machine, loading an image to the device, resetting the mirrors to the their rest state etc. \layout Standard \paragraph_spacing double As a multi-thread \begin_inset Foot collapsed false \layout Standard A thread in computer science is short for a thread of execution. Threads are a way for a program to split itself into two or more simultaneously running tasks. This multi-threading generally occurs by time slicing (where a single processor switches between different threads. We use threads to switch between socket operations and display control of the device.The server starts a thread to display the device and another thread to receive calls from the client. \end_inset windows application the server application can simultaneously control the device as well as function as a socket based application receptive to incoming connection requests. When a request for a connection is made by a client (remote user), the server, ensuring that there are no present users of the device, permits the connection. After the connection is established successfully the server receives preprocess ed images sent in by the client and starts a separate thread which loads the images to the device. The images are continually displayed on the device till the next image is received by the server and the entire process is repeated. The server application has a GUI (graphs user interface) and features which include logging the IP address and date/time of connections as well as automatically generated error messages in case of failures. These features provide for easy maintenance and further software development of the application. \layout Standard \paragraph_spacing double We also developed a client application for remote users to communicate with the device, remotely over the ethernet. This remote access is critical for the functioning of the device as the actual DMD is in the beam path of a class IV laser and cannot be accessed during beam operations. The client application also has a graphic interface for ease of use. The client application preprocessors any image file that needs to be loaded into the DMD. This preprocessing ensures quick uploading of the images to the DMD which is essential for the time critical part of experiments. To completely automate the process of sending a series of images to the device at regular user defined intervals a dynamic link library (DLL) \begin_inset Foot collapsed false \layout Standard A dynamic link library is a windows version of a shared library, that contains subroutines to be linked to a binary executable at run time. This allows sharing of code between many applications. \end_inset is created. This DLL is called from a mathcad sheet, completely automates the process of establishing connection, communicating and sending images to the server. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/SoftwareScheme1.eps display none scale 65 clip rotateOrigin center \end_inset \layout Caption Software scheme to control the DMD. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section \paragraph_spacing double Experimental Setup \begin_inset LatexCommand \label{sec:Ch5_ExperimentSetup} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/ExperimentalScheme1.eps display none \end_inset \layout Caption This figure shows the schematic sketch of the experiment conducted at the ATF. \begin_inset LatexCommand \label{cap:Ch5_ExpSetUp} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The experiment was conducted at the Accelerator Test Facility [ \begin_inset LatexCommand \ref{ATFOverview} \end_inset ] at BNL and a schematic of the set up is shown in \begin_inset LatexCommand \ref{cap:Ch5_ExpSetUp} \end_inset . The beam from the Nd:Yag laser has wavelength of 1024 \begin_inset Formula $nm$ \end_inset . This beam is incident upon a frequency doubling crystal, called a second harmonic generation (SHG) crystal. Such a crystal generates a wave-packet of double the frequency than the incident light. The resulting wavelength at 532 \begin_inset Formula $nm$ \end_inset is sent to the DMD device. The DMD, being a commercial device works in the visible range of the electromag netic spectrum and is thus placed after the first SHG. The reflected light from the DMD is sent to a second SHG crystal which produces an output beam having a wavelength of 266 \begin_inset Formula $nm$ \end_inset . This UV light at 266 \begin_inset Formula $nm$ \end_inset wavelength (4.2 eV photon energy) is a requirement for the Mg photo-cathode (work function is 3.2 eV) to emit electrons. The 266 \begin_inset Formula $nm$ \end_inset wavelength beam is then carried through 20 \begin_inset Formula $m$ \end_inset of transport line in vacuum and finally irradiates the cathode surface. A beam splitter is placed just before the cathode. A part of the beam is split to form the image of the cathode[ \begin_inset LatexCommand \ref{optics} \end_inset ]. The laser energy irradiating the cathode is measured by a calibrated pick off from a joule meter which receives a second surface reflection from the beam splitter. \layout Section \paragraph_spacing double New Quantum Efficiency Measurement Technique \begin_inset LatexCommand \label{sec:Ch5_New-QETechnique} \end_inset \layout Standard \paragraph_spacing double Quantum Efficiency [ \begin_inset LatexCommand \ref{QE} \end_inset ] of a photo-cathode is an important characteristic that determines the quality of the photo-cathode. The digital mirror device(DMD) was used to measure the quantum efficiency of the photo-cathode at ATF. An image was uploaded into the device such that only a sub-array of mirrors, 64X64 out of the entire range of 1024X768, are in ON position. The mirrors that are in ON state reflect light onto a small localised section of the photo-cathode. This illuminates a small section of the cathode. Electrons are emitted only from the irradiated section of the photo-cathode as parts of the photo-cathode in darkness do not emit electrons. Using the DMD the entire surface of the photo-cathode is scanned by turning different sub-arrays of of the mirrors which in turn illuminates different sections of the device. Thus loading images on the DMD such that a different subset set of mirrors reflects light onto different parts of the cathode gives us an effective way of scanning through the surface of the cathode with very good resolution. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/QE1.eps display none \end_inset \layout Caption Surface plots from QE measurements. \begin_inset LatexCommand \label{cap:Ch5_QE1} \end_inset \layout Standard (a)shows the charge measurement as a surface plot obtained by scanning the different parts of the ATF photo-cathode surface (b) Shows the surface plot of the incident laser energy measurement. (c) The surface plot of the QE map is plotted. In all the three plots the x and the y axis have number of elements scanning the surface of the cathode. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The scanning technique in place we can measure the quantum efficiency of the photo-cathode by measuring the emitted charge and the laser energy incident upon the photo-cathode surface. The charge is measured by a Faraday cup. The Faraday cup is an isolated electrode, designed to stop and capture electron beam charge. To measure the beam charge an oscilloscope is used to measure the current across the Faraday cup when an electron bunch hits it. The area under the current signal, calculated by the oscilloscope's numerical analysis hardware gives the charge. The energy is measured by the joule-meter placed before the cathode as discussed in [ \begin_inset LatexCommand \ref{sec:Ch5_ExperimentSetup} \end_inset ]. The entire process of measuring charge and laser energy is automated at ATF by building a customised Mathcad dynamic link library (DLL) which communica tes with the oscilloscope over a gpib \begin_inset Foot collapsed false \layout Standard General Purpose Instrumentation Bus (GPIB) is the IEEE Standard Digital Interface for Programmable Instrumentation. \end_inset port. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/QE_demo.eps display none clip \end_inset \layout Caption QE Maps of the ATF photo-cathode. \begin_inset LatexCommand \label{cap:Ch5_QEAndDeviation} \end_inset \layout Standard (a) Shows the QE map of the ATF photo-cathode as a three dimensional surface plot. The x and the y axis show a grid size of 5 by 5 and the QE in % is given along the z axis. The color is proportional to the surface height in RGB. (b) Shows the spatial uniformity of of QE map in % estimated by the standard deviation from the mean value of the map. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double Measured charge q is easily converted to number of electrons as \begin_inset Formula $q=N.e$ \end_inset where \begin_inset Formula $N$ \end_inset is the number of electrons and \begin_inset Formula $e$ \end_inset is the charge of an electron. The laser energy ( \begin_inset Formula $E$ \end_inset ) can be converted to number of photons as \begin_inset Formula $E=M.h\nu$ \end_inset where \begin_inset Formula $M$ \end_inset is the number of photons, \begin_inset Formula $h\nu$ \end_inset is the photon energy where \begin_inset Formula $h$ \end_inset is the Planck's constant and \begin_inset Formula $\nu$ \end_inset is the frequency of incident light. For the 4th harmonic of the ATF Nd:Yag laser \begin_inset Formula $(\lambda=266nm)$ \end_inset that is incident upon the photo-cathode, the photon energy is about \begin_inset Formula $4.2$ \end_inset eV. Thus dimensionless QE can be calculated as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} QE=\frac{Charge/electronCharge}{laserEnergy/h\nu}=\frac{N(\textnormal{number of electrons emitted)}}{M(\textnormal{number of photons incident)}}\end{equation} \end_inset \layout Standard \paragraph_spacing double It is standard practise to express QE in terms of % and we do the same through out this thesis. A representative measurement is shown in figure \begin_inset LatexCommand \ref{cap:Ch5_QE1} \end_inset , figure \begin_inset LatexCommand \ref{cap:Ch5_QE1} \end_inset (a) and figure \begin_inset LatexCommand \ref{cap:Ch5_QE1} \end_inset (b) show the surface plot of the measured charge and incident energy respective ly when a nine by nine raster scan is performed over the photo-cathode surface area. Figure \begin_inset LatexCommand \ref{cap:Ch5_QE1} \end_inset (c) shows the surface plot of the QE of the photo-cathode, where the QE values are in %. We also study the variation of the QE map over the surface of the photo-cathode as the uniformity of the photo-cathode is an important figure of merit to consider for producing uniform electron beams. In figure we show another representative graph of the QE map of the photo-catho de and a map depicting the uniformity of the photo-cathode. The uniformity is calculated by taking the standard deviation (in %) of the QE measurements during a single measurement of scanning the photo-cathode with nine by nine illuminated spots. \layout Section \paragraph_spacing double Laser Cleaning of Photo-cathode \begin_inset LatexCommand \label{sec:Ch5_Laser-Cleaning-Cathode} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/QE_compare1.eps lyxscale 20 display none clip \end_inset \layout Caption Comparison of QE before and after laser cleaning of ATF photo-cathode. \begin_inset LatexCommand \label{cap:Ch5_QE_compare} \end_inset \layout Standard This figure compares the QE maps of the ATF photo-cathode before and after laser cleaning procedure is applied to the cathode surface. In (a), the QE map before the surface is cleaned is presented. The mean QE is found to be 0.01% and the median value is found to be 0.01% . In (b) the QE map of the photo-cathode is shown after it has been cleaned by focusing laser light on it. The mean and median values of QE are found to be 0.09%. The experimental error in the measurements is about 6%. Comparison of the two plots shows that the QE increases by a factor of 9 by the cleaning procedure that is applied. \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/QE3.eps display none clip \end_inset \layout Caption Spatial variation in QE before and after laser cleaning of ATF photo-cathode. \begin_inset LatexCommand \label{cap:Ch5_QE_SpatialDevCompare} \end_inset \layout Standard (a),the spatial variation of the QE map before the surface is cleaned is presented.The variation is found to be 18% (b) the QE map of the photo-cathode is shown after it has been cleaned by focusing laser light on it. The QE variation across the surface of the photo-cathode is found to be about 19% with in experimental relative error of about 6%. Comparison of the two plots shows that the spatial uniformity does not improve much with the cleaning procedure . \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double Breakdowns and arcing in the RF gun tends to reduce the QE of a photo-cathode. To restore the QE of the photo-cathode a technique of cleaning the surface of the cathode has been developed at at the ATF \begin_inset LatexCommand \cite{key-27} \end_inset . In this procedure the cathode is slowly scanned with a high peak power laser maintaining an RF field of 67 MV/m on the cathode. At each scan site the laser energy is adjusted till there is an explosive emission from the cathode for three consecutive shots. This procedure is believed to improve the QE, both in terms of numerical value and uniformity, of the photo-cathode by ablating the contaminated surface of the photo-cathode. \layout Subsection \paragraph_spacing double Results \begin_inset LatexCommand \label{sub:Ch5_Results} \end_inset \layout Standard \paragraph_spacing double We measured the QE and its spatial deviation both before and after the cleaning procedure had been applied and present the results in figure \begin_inset LatexCommand \ref{cap:Ch5_QE_compare} \end_inset and \begin_inset LatexCommand \ref{cap:Ch5_QE_SpatialDevCompare} \end_inset respectively. Though the quantum efficiency improves by a factor of 9, from 0.01% before cleaning to 0.09% after cleaning, our results show that there is nearly no change in the uniformity of the cathode surface. The spatial deviation is found to be 18% before the cleaning procedure is applied and about 19% after the cleaning procedure and within experimental error of about 6%, it is indicative of the uniformity not being affected by the laser cleaning procedure. \layout Section \paragraph_spacing double Summary \layout Standard \paragraph_spacing double In this chapter we discuss the the new technique developed at ATF to measure the quantum efficiency (QE) of a photo-cathode using a digital mirror device to scan the surface of the photo-cathode. Apart from simply measuring the QE, the technique also gives us a useful tool to study the spatial deviation of a photo-cathode surface with much greater resolution. This technique was successfully applied to measure the QE of the photo-cathode at ATF. We found that the results from our technique agree very well with QE measuremen ts from traditional methods. The QE measurements were done both before and after the laser cleaning method was applied to the photo-cathode. The cleaning of the cathode surface yielded much higher QE as was expected. Contrary to expectations the study of the spatial profiles showed that the uniformity of the cathode surface does not improve much by the cleaning procedure. This technique also brings down the time required to get a QE map of the photo-cathode surface from a few hours to about 20 minutes for a nine by nine raster scan. \layout Chapter Emittance-Theory, Measurements and Simulations \layout Section Introduction \layout Standard \paragraph_spacing double In this chapter, we discuss experiments and simulations done to investigate the effect of non-uniform laser distributions on the emittance of an electron beam. We start by discussing the relevant theoretical framework to understand the electron beam evolution in terms of its envelope. The envelope of a beam, hiding the details of individual particle motion, gives us the aggregate behaviour of the beam.The beam envelope, its evolution and the consequent evolution of the emittance is discussed in Section \begin_inset LatexCommand \ref{sec:TheoryEnvelope} \end_inset . The theoretical models, though they provide insight into electron beam evolution, are restricted to the linear regime of forces. The effect of non-linearities, must be investigated experimentally or by numerical simulations. We implement both approaches to study the effect of non-linearities on the electron beam emittance. The DMD as a spatial light modulator [ \begin_inset LatexCommand \ref{sub:DMD-asSLM} \end_inset ] is used to create controlled distortions in the transverse distribution of the drive laser beam. This distorted laser beam is then incident upon the cathode surface and the emittance of the electron beam is measured. Such a measurement is described in section \begin_inset LatexCommand \ref{sub:Measurement-of-Emittance} \end_inset . In section \begin_inset LatexCommand \ref{sec:ExperimentsEmittance} \end_inset we present results of experiments wherein we measured emittance of the electron beam, at the ATF, by systematically introducing non-uniformity in the laser beam. Our results confirm that non-uniform laser distributions increase the emittance of an electron beam. We also find that this effect is mitigated by fine-graining of the distortions. To corroborate our experimental findings we simulate the process by generating particle distributions which emulate the laser distributions used for experimen ts. These simulations are done by using the beam simulations code PARMELA and results from these simulations are shown in section \begin_inset LatexCommand \ref{sec:Parmela-Simulations} \end_inset . We find good agreement between the depicted behavior in experiments and the simulated results as illustrated in figure \begin_inset LatexCommand \ref{cap:Ch6_F21compare} \end_inset . Finally in section \begin_inset LatexCommand \ref{sec:PhysicsOfResults} \end_inset we present a physical understanding of our findings. \layout Section \paragraph_spacing double Theory \begin_inset LatexCommand \label{sec:TheoryEnvelope} \end_inset \layout Standard \paragraph_spacing double We have introduced the concept of electron beam emittance as a figure of merit for an electron beam [ \begin_inset LatexCommand \ref{Emittance} \end_inset ]. To recapitulate, emittance is defined as \begin_inset Formula \begin{eqnarray*} \tilde{\epsilon_{x}} & = & \left(\langle x^{2}\rangle\langle x'^{2}\rangle-\langle xx'\rangle^{2}\right)^{1/2}\end{eqnarray*} \end_inset \layout Standard In terms of second moments \begin_inset Formula \begin{equation} \epsilon_{x}=\sigma_{x}^{2}\sigma_{x'}^{2}-\sigma_{xx'}^{2}\label{eq:Ch6_1}\end{equation} \end_inset \layout Standard Pedagogy dictates that we discuss the evolution of the electron beam to gain an understanding into emittance, its growth and compensation of this growth, before we discuss emittance measurements. To follow the electron beam evolution which is an ensemble of large number of particles, and thus analytically intractable, we follow the envelope of the beam instead. The envelope of a beam can be visualised as \emph on \emph default the trace space ellipse [ \begin_inset LatexCommand \ref{trace_space} \end_inset ] encompassing the particle distribution. This beam envelope, gives us information \emph on \emph default about \emph on \emph default the extent of the distribution \emph on . \emph default We derive the envelope equation, as a differential equation governing the evolution of the second moments in terms of the transverse trace space variables x and y. We derive the equation for x, remembering that a similar equation may be written down for the y-y' trace space. \layout Standard We begin by taking the first derivative of the rms beam size. \layout Standard \begin_inset Formula \begin{eqnarray} \frac{d\sigma_{x}}{dz} & = & \frac{d}{dz}\sqrt{\left\langle x^{2}\right\rangle }=\frac{1}{2\sigma_{x}}\frac{d}{dz}\langle x^{2}\rangle\label{eq:Ch6_2}\\ \equiv & \frac{1}{2\sigma_{x}} & \frac{d}{dz}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x^{2}f(x,x')dxdx'\label{eq:Ch6_3}\\ \equiv & \frac{1}{\sigma_{x}} & \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xx'f(x,x')dxdx'=\frac{\sigma_{xx'}}{\sigma_{x}}\label{Ch6_4}\end{eqnarray} \end_inset \layout Standard The second derivative of the rms beam size is given by \begin_inset Formula \begin{eqnarray} \frac{d^{2}\sigma_{x}}{dz^{2}} & = & \frac{d}{dz}\frac{\sigma_{xx'}}{\sigma_{x}}=\sigma_{xx'}\frac{d}{dz}\left(\frac{1}{\sigma_{x}}\right)+\frac{1}{\sigma_{x}}\frac{d}{dz}\left(\sigma_{xx'}\right)\label{eq:Ch6_5}\\ \frac{d^{2}\sigma_{x}}{dz^{2}} & = & \sigma_{xx'}\left(\frac{-1}{\sigma_{x}^{2}}\frac{d}{dz}\sigma_{x}\right)+\frac{1}{\sigma_{x}}\frac{d}{dz}\left(\sigma_{xx'}\right)\label{eq:Ch6_6}\end{eqnarray} \end_inset Substituting from equation \begin_inset LatexCommand \ref{eq:Ch6_3} \end_inset into equation \begin_inset LatexCommand \ref{eq:Ch6_5} \end_inset \begin_inset Formula \begin{eqnarray} \frac{d^{2}\sigma_{x}}{dz^{2}} & =\frac{1}{\sigma_{x}} & \frac{d}{dz}\left(\sigma_{xx'}\right)-\sigma_{xx'}\left(\frac{1}{\sigma_{x}^{2}}\left(\frac{\sigma_{xx'}}{\sigma_{x}}\right)\right)\label{eq:Ch6_7}\\ \rightarrow\frac{d^{2}\sigma_{x}}{dz^{2}} & = & \frac{1}{\sigma_{x}}\frac{d}{dz}\left(\sigma_{xx'}\right)-\frac{\sigma_{xx'}^{2}}{\sigma_{x}^{3}}\label{eq:Ch6_8}\\ \rightarrow\frac{d^{2}\sigma_{x}}{dz^{2}} & = & \frac{1}{\sigma_{x}}\frac{d}{dz}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xx'f(x,x')dxdx'-\frac{\sigma_{xx'}}{\sigma_{x}^{3}}\label{eq:Ch6_9}\\ \rightarrow\frac{d^{2}\sigma_{x}}{dz^{2}} & = & \frac{1}{\sigma_{x}}\int_{-\infty}^{\infty}\frac{dx}{dz}x'f(x,x')dxdx'+\int_{-\infty}^{\infty}x\frac{dx'}{dz}xf(x,x')dxdx'-\frac{\sigma_{xx'}}{\sigma_{x}^{3}}\label{eq:Ch6_10}\end{eqnarray} \end_inset \layout Standard By definition, \begin_inset Formula $\frac{dx}{dz}=x'$ \end_inset and \begin_inset Formula $\frac{dx'}{dz}=x''$ \end_inset .Thus equation \begin_inset LatexCommand \ref{eq:Ch6_10} \end_inset becomes \begin_inset Formula \begin{eqnarray} \frac{d^{2}\sigma_{x}}{dz^{2}} & = & \frac{1}{\sigma_{x}}\int_{-\infty}^{\infty}x'x'f(x,x')dxdx'+\int_{-\infty}^{\infty}xx''f(x,x')dxdx'-\frac{\sigma_{xx'}}{\sigma_{x}^{3}}\label{eq:Ch6_11}\end{eqnarray} \end_inset \layout Standard The first term in equation \begin_inset LatexCommand \ref{eq:Ch6_11} \end_inset is by definition the square of second moment of \begin_inset Formula $x'$ \end_inset and thus can be replaced by \begin_inset Formula $\sigma_{x'}^{2}$ \end_inset and the second term is \begin_inset Formula $\langle xx'\rangle$ \end_inset . \begin_inset Formula \begin{equation} \frac{d^{2}\sigma_{x}}{dz^{2}}=\frac{\sigma_{x'}^{2}+\langle xx''\rangle}{\sigma_{x}}-\frac{\sigma_{xx'}}{\sigma_{x}^{3}}\label{eq:Ch6_12}\end{equation} \end_inset \layout Standard Rearranging the terms on right hand side of equation \begin_inset LatexCommand \ref{eq:Ch6_12} \end_inset we have, \begin_inset Formula \begin{equation} \sigma_{x}''=\frac{\sigma_{x}^{2}\sigma_{x'}^{2}-\sigma_{xx'}^{2}}{(\sigma_{x})^{3}}-\frac{\langle xx''\rangle}{\sigma_{x}}\label{eq:Ch6_13}\end{equation} \end_inset \layout Standard We know that \begin_inset Formula $\epsilon_{x}$ \end_inset is \begin_inset Formula \begin{equation} \epsilon_{x,rms}=\sigma_{x}^{2}\sigma_{x'}^{2}-\sigma_{xx'}^{2}\label{eq:Ch6_14}\end{equation} \end_inset \layout Standard Hence equation \begin_inset LatexCommand \ref{eq:Ch6_13} \end_inset maybe written as \begin_inset Formula \begin{equation} \sigma''_{x}=\frac{\epsilon_{x,rms}}{(\sigma_{x})^{3}}-\frac{\langle xx''\rangle}{\sigma_{x}}\label{eq:Ch6_15}\end{equation} \end_inset \layout Standard Under linear transport conditions, the general form of the equation of motion is \begin_inset Formula \begin{equation} x''=-\kappa_{x}^{2}x\label{eq:Ch6_16}\end{equation} \end_inset \layout Standard Taking an average of the equation \begin_inset LatexCommand \ref{eq:Ch6_16} \end_inset , we have \layout Standard \begin_inset Formula \begin{equation} \langle xx''\rangle=-\kappa_{x}^{2}\langle x^{2}\rangle\label{eq:Ch6_17}\end{equation} \end_inset \layout Standard Equation 6.13 can now be summarised as \begin_inset Formula \begin{equation} \mathbf{\mathbf{\sigma''_{x}+\kappa_{x}^{2}\sigma_{x}=\frac{\left(\epsilon_{x,rms}\right)^{2}}{\sigma_{x,}^{3}}}}\label{eq:Ch6_18EnvelopeEqnSimple}\end{equation} \end_inset \layout Standard Equation \begin_inset LatexCommand \ref{eq:Ch6_18EnvelopeEqnSimple} \end_inset is the rms envelope equation, where \begin_inset Formula $\kappa_{x}$ \end_inset maybe positive, negative or vanishing and the right hand side is the force like term parametrised by the emittance. \layout Subsection Acceleration Term \layout Standard In electron sources, there is a strong acceleration of the electron-beam, that effect may be included by adding an extra term to equation \begin_inset LatexCommand \ref{eq:Ch6_18EnvelopeEqnSimple} \end_inset . This term accounts for the instantaneous spatial rate of momentum change \begin_inset Formula $(p_{o})'=(\beta_{o}\gamma_{o})'m_{o}c$ \end_inset . From equation \begin_inset LatexCommand \ref{eq:Ch6_8} \end_inset we can write \begin_inset Formula \begin{equation} \frac{d^{2}\sigma_{x}}{dz^{2}}=\frac{d}{dz}\frac{\sigma_{xx'}}{\sigma_{x}}=\frac{1}{\sigma_{x}}\frac{d}{dz}\left(\sigma_{xx'}\right)-\frac{\sigma_{xx'}^{2}}{\sigma_{x}^{3}}\label{eq:Ch6_19}\end{equation} \end_inset \layout Standard \begin_inset Formula \begin{equation} =\frac{1}{\sigma_{x}}\left[\sigma_{x'}^{2}-\kappa_{x}^{2}\sigma_{x}^{2}-\frac{(\beta_{o}\gamma_{o})'}{\beta_{o}\gamma_{o}}\sigma_{xx'}\right]-\frac{\sigma_{xx'}^{2}}{\sigma_{x}^{3}}\label{eq:Ch6_20}\end{equation} \end_inset \layout Standard We introduce normalised emittance to write the equation in standard form \layout Standard \begin_inset Formula \begin{equation} \frac{d^{2}\sigma_{x}}{dz^{2}}+\frac{\left(\beta\gamma\right)^{'}}{\left(\beta\gamma\right)}\frac{d\sigma_{x}}{dz}+\kappa_{x}^{2}\sigma_{x}=\frac{\epsilon_{n,x}^{2}}{\left(\beta\gamma\right)^{2}\sigma_{x}^{3}}\label{eq:Ch6_21}\end{equation} \end_inset \layout Standard where normalised emittance is \begin_inset Formula \begin{equation} \epsilon_{n,x}\equiv\beta_{o}\gamma_{o}\epsilon_{x,rms}\label{eq:Ch6_22}\end{equation} \end_inset \layout Subsection Space Charge Term \layout Standard We now include the effect on the envelope equation, due to space charge forces. Space charge forces are the self fields of a charged particle which may cause spreading or self constriction of the beam. Let us focus on radial forces in paraxial approximation i.e \begin_inset Formula $|x'|<1$ \end_inset , \begin_inset Formula $|y'|<1$ \end_inset and \begin_inset Formula $v_{z}\approx|\vec{v}|=v$ \end_inset . There is an electrostatic outward force and an inward force arising from the particle motion in the self magnetic field. Further for constant charge density \begin_inset Formula $\rho$ \end_inset as well as current density \begin_inset Formula $\vec{J}$ \end_inset we write for \begin_inset Formula $0\leq r\leq a$ \end_inset \layout Standard \begin_inset Formula \begin{eqnarray} J_{z} & = & J=\frac{I}{a^{2}\pi}.\label{eq:Ch6_23}\\ \rho & = & \frac{I}{a^{2}\pi v},\label{eq:ch6_24}\end{eqnarray} \end_inset \layout Standard and \begin_inset Formula $J=0$ \end_inset and \begin_inset Formula $\rho=0$ \end_inset for \begin_inset Formula $r>a$ \end_inset , where a is the radius of the beam. \layout Standard The radial electric field at a distance r is given by Gauss's law applied to a cylinder of radius r and unit length in z direction \begin_inset Formula \begin{eqnarray} \int\epsilon_{o} & \vec{E\cdot}\vec{dS} & =\int\rho dV;\label{eq:Ch6_25}\end{eqnarray} \end_inset \layout Standard which gives \begin_inset Formula \begin{equation} E_{r}=\frac{\rho_{o}}{2\epsilon_{o}}=\frac{Ir}{2\pi\epsilon_{o}a^{2}v}.\label{eq:Ch6_26}\end{equation} \end_inset \layout Standard The magnetic field is defined by Ampere's law as \begin_inset Formula \begin{equation} \int\vec{B}\cdot d\vec{l}=\mu_{o}\int\vec{J}\cdot\vec{dS}\label{eq:Ch6_27}\end{equation} \end_inset \layout Standard which gives \begin_inset Formula \begin{equation} B_{\phi}=\mu_{o}\frac{Ir}{2\pi a^{2}}.\label{eq:Ch6_28}\end{equation} \end_inset \layout Standard The radial force equation can thus be written,under the assumption \begin_inset Formula $v\approx v_{z}$ \end_inset and that \begin_inset Formula $\gamma$ \end_inset remains constant as \begin_inset Formula \begin{eqnarray} \frac{d}{dt}(\gamma m\dot{r}) & = & \gamma m\ddot{r}=eE_{r}-ev_{z}B_{\phi}\label{eq:Ch6_29}\\ \gamma m\ddot{r} & = & eE_{r}-evB_{\phi}\label{eq:Ch6_29}\end{eqnarray} \end_inset \layout Standard Substituting for \begin_inset Formula $E_{r}$ \end_inset and \begin_inset Formula $B_{\phi}$ \end_inset we can write the radial equation as \layout Standard \begin_inset Formula \begin{eqnarray} \gamma m\ddot{r} & = & \frac{eIr}{2\pi\epsilon_{o}a^{2}v}+\mu_{o}\frac{eIr}{2\pi a^{2}}v\label{eq:Ch6_30}\\ \gamma m\ddot{r} & = & \frac{eIr}{2\pi\epsilon_{o}a^{2}\beta c}(1-\beta^{2})\label{eq:Ch6_31}\end{eqnarray} \end_inset \layout Standard substituting \begin_inset Formula \begin{equation} \ddot{r}=v_{z}^{2}\frac{d^{2}r}{dz^{2}}=\beta^{2}c^{2}r''\label{eq:Ch6_32}\end{equation} \end_inset \layout Standard We write equation \begin_inset LatexCommand \ref{eq:Ch6_32} \end_inset as \begin_inset Formula \begin{equation} r''=\frac{eIr}{2\pi\epsilon_{o}a^{2}mc^{3}\beta^{3}\gamma^{3}}\label{eq:Ch6_33}\end{equation} \end_inset \begin_inset Formula \begin{equation} r''=\frac{Ir}{2I_{o}a^{2}(\beta\gamma)^{3}}\label{eq:Ch6_34}\end{equation} \end_inset where \begin_inset Formula $I_{o}=\frac{4\pi\epsilon_{o}mc^{3}}{e}\approx17KA$ \end_inset for electrons and is called the \begin_inset ERT status Open \layout Standard Alfv \backslash 'en current. \end_inset \layout Standard Without loss of generality, we can replace the radial coordinate \begin_inset Formula $r$ \end_inset by the rms beam size \begin_inset Formula $\sigma$ \end_inset to ascertain the effect of the space charge force on the envelope equation \begin_inset Formula \begin{equation} \sigma''_{spacecharge}=\frac{I\sigma}{2I_{o}a^{2}(\beta\gamma)^{3}}\label{eq:Ch6_35}\end{equation} \end_inset \layout Standard Under conditions of laminar flow,the trajectories of all particles are similar and specifically the particle at \begin_inset Formula $r=a=\sigma$ \end_inset , always remains at the boundary of the beam. The space charge force thus scales as \begin_inset Formula \begin{equation} \sigma''_{spacecharge}=\frac{I}{2I_{o}\sigma(\beta\gamma)^{3}}\label{eq:Ch6_37}\end{equation} \end_inset \layout Standard Thus the envelope equation can be written as \begin_inset Formula \begin{equation} \frac{d^{2}\sigma}{dz^{2}}+\left[\frac{\left(\beta\gamma\right)^{'}}{\left(\beta\gamma\right)}\frac{d\sigma}{dz}\right]+\left[\kappa\sigma\right]=\left[\frac{I}{2I_{o}\sigma(\beta\gamma)^{3}}\right]+\left[\frac{\epsilon_{n}^{2}}{\left(\beta\gamma\right)^{2}\sigma^{3}}\right].\label{eq:Ch6_35EnvelopeFinal}\end{equation} \end_inset \layout Standard \begin_inset LatexCommand \ref{eq:Ch6_35} \end_inset dictates the evolution of the envelope of the electron beam, where \begin_inset Formula $\frac{\left(\beta\gamma\right)^{'}}{\left(\beta\gamma\right)}$ \end_inset is effect of acceleration, and \begin_inset Formula $\kappa$ \end_inset is the focusing strength of the transport elements. The first term on the right hand side of the equation is the consequence of the space charge forces and the second term on the right hand side is the normalised rms emittance which exerts an outward force on the beam envelope. \layout Subsection Emittance Evolution \begin_inset LatexCommand \label{sub:Emittance-Evolution} \end_inset \layout Standard \paragraph_spacing double In order to understand the mechanism that causes projected emittance growth and its possible growth compensation, we discuss a model by Serafini and Rosenzweig \begin_inset LatexCommand \cite{key-66} \end_inset . This model views the projected emittance as arising from the phase space dynamics of each longitudinal slice of the beam. The beam is considered to be cold (mono-energetic), laminar (electron trajector ies do not cross) and space charge dominated plasma evolving under linear external forces. This model ignores the effect of acceleration and transverse motion due to high gradient RF fields. These approximations are sufficient to reveal the fundamental dynamics of the emittance evolution which are only slightly modified in real accelerator s. \layout Standard From equation \begin_inset LatexCommand \ref{eq:Ch6_35EnvelopeFinal} \end_inset , ignoring the acceleration term, the rms envelope equation for a cylindrically symmetric, space-charge dominated, charged particle beam in a focusing channel of constant strength is \begin_inset Formula \begin{eqnarray} \sigma'' & +\kappa^{2}\sigma & =\frac{I}{2I_{o}(\beta\gamma)^{3}\sigma}+\frac{\epsilon_{n}^{2}}{\left(\beta\gamma\right)^{2}\sigma^{3}}\label{eq:Ch6_36EnvelopeSpaceharge}\end{eqnarray} \end_inset \layout Standard The last term on the right hand side of equation \begin_inset LatexCommand \ref{eq:Ch6_36EnvelopeSpaceharge} \end_inset , represents the contributions to envelope evolution arising from the finite thermal emittance of the beam as well as nonlinear effects. This term is small in very bright sources and may be ignored for space-charge dominated beams. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/ComensationPhaseSpace.eps display none \end_inset \layout Caption Projected Phase Space. \begin_inset LatexCommand \label{cap:Ch6_1Projected-Phase-Space} \end_inset \layout Standard (a) shows the longitudinal slices in an exaggerated size of electron bunch. (b) shows the trace space corresponding the slice marked edge in (a). (c) shows the trace space for the slice marked core in (a). \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard We now consider longitudinal slices of the beam, of infinitesimal length \begin_inset Formula $\zeta=z-ct$ \end_inset as shown in figure \begin_inset LatexCommand \ref{cap:Ch6_1Projected-Phase-Space} \end_inset . In the co-moving frame the electron bunch length is significantly larger than its radius. As a result, the transverse force is determined by the local current \begin_inset Formula $I(\zeta)$ \end_inset and is largely independent of the rest of the bunch provided \begin_inset Formula $I(\zeta)$ \end_inset is a smooth function of \begin_inset Formula $\zeta$ \end_inset , i.e. \begin_inset Formula $|\frac{\partial I(\zeta)}{\partial\zeta}|<\frac{I}{\sigma}$ \end_inset . The normalized emittance of such a slice is defined as \begin_inset Formula \begin{equation} \epsilon_{n,slice}\equiv\frac{\beta\gamma}{2}\sqrt{\langle r^{2}\rangle_{\zeta}\langle r'^{2}\rangle_{\zeta}-\langle rr'\rangle_{\zeta}^{2}}\label{eq:Ch6_37}\end{equation} \end_inset where subscript \begin_inset Formula $\zeta$ \end_inset defines average being taken over a slice. \layout Standard The space charge term is also generalised to include an explicit dependence on longitudinal position by \begin_inset Formula $I\rightarrow Ig(\zeta)$ \end_inset , where I is the maximum current in the beam. \begin_inset Formula $g(\zeta)$ \end_inset is a geometrical factor in the rest frame of the electrons. The local equilibrium solution ( \begin_inset Formula $\sigma''=0$ \end_inset ) for equation for a slice \begin_inset Formula $\zeta$ \end_inset is \begin_inset Formula \begin{equation} \sigma_{eq}(g(\zeta))=\left(\frac{Ig(\zeta)}{2I_{o}(\beta\gamma)^{3}\kappa}\right)^{1/2}\label{eq:Ch6_38}\end{equation} \end_inset \layout Standard Linearising equation \begin_inset LatexCommand \ref{eq:Ch6_36EnvelopeSpaceharge} \end_inset about this equilibrium position for small envelope deviation, i.e. \begin_inset Formula $|\delta\sigma|\ll|\sigma_{eq}|$ \end_inset \begin_inset Formula \begin{equation} \delta\sigma^{''}(\zeta)+\left[\kappa+\frac{Ig(\zeta)}{2I_{o}(\beta\gamma)^{3}\sigma(g(\zeta))_{eq}^{2}}\right]\delta\sigma(\zeta)=0,\label{eq:Ch6_38}\end{equation} \end_inset or \begin_inset Formula \begin{equation} \mathbf{\mathbf{\delta\sigma^{''}(\zeta)+2\kappa\delta\sigma(\zeta)=0}}.\label{eq:Ch6_39EnvelopeSimplified}\end{equation} \end_inset \layout Standard Equation \begin_inset LatexCommand \ref{eq:Ch6_39EnvelopeSimplified} \end_inset gives the oscillation frequencies that depend on external focusing but are independent of the beam current. Thus all envelope oscillations have the same frequency, given the same external focusing strength. The formal solution to equation \begin_inset LatexCommand \ref{eq:Ch6_39EnvelopeSimplified} \end_inset for a given beam size \begin_inset Formula $\sigma_{o}$ \end_inset may be written as \begin_inset Formula \begin{eqnarray} \sigma(z,\zeta) & = & \sigma_{eq}(g(\zeta))+\left(\sigma_{o}-\sigma_{eq}(g(\zeta)\right)\cos\left(\sqrt{2\kappa}z\right)\label{eq:Ch640envelopeSoln1}\\ \sigma^{'}(z) & = & -2\sqrt{\kappa}\left(\sigma_{0}-\sigma_{eq}(g(\zeta))\right)\sin\left(\sqrt{2\kappa}z\right).\label{eq:Ch6_41envelopeSoln2}\end{eqnarray} \end_inset \layout Standard Since emittance is \begin_inset Formula \begin{equation} \epsilon_{projected}(z)=\sqrt{\langle\sigma^{2}\rangle_{\zeta}\langle\sigma'\rangle_{\zeta}-\langle\sigma\sigma'\rangle_{\zeta}}\label{eq:Emittanceredefine}\end{equation} \end_inset \layout Standard Equation \begin_inset LatexCommand \ref{eq:Emittanceredefine} \end_inset shows that the rms emittance of the beam also oscillates and periodically reaches maxima and minima. For photo-injectors the beam is allowed to go through only one minimum of the envelope oscillation and then the minimum value of emittance is frozen by accelerating the beam to a regime where the space charge forces become negligible. The scheme is discussed in the next section. There exists a particular solution to the envelope equation, known as the invariant envelope for which the final state of the beam has vanishing emittance under the linear force regime. Compensation of emittance is achieved by matching the beam to this invariant envelope at the end of a focusing solenoid. \layout Subsection Emittance Compensation \begin_inset LatexCommand \label{sub:Emittance-Compensation} \end_inset \layout Standard We now discuss qualitatively the scheme of emittance compensation as implemented in photo-injectors. \begin_inset LatexCommand \label{sub:Emittance-Compensation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Emittance_CompensationVl.eps display none scale 75 clip \end_inset \layout Caption Transverse Phase Space plots: (a) Initial Condition; (b) After Some drift; (c) immediately after lens; (d) after additional drift behind lens. \begin_inset LatexCommand \label{cap:Ch6_2Emittance Compensation} \end_inset \layout Standard picture courtesy V.N. Litvinenko \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The evolution of emittance compensation can be described in four steps shown in figure \begin_inset LatexCommand \ref{cap:Ch6_2Emittance Compensation} \end_inset . The initial beam with a uniform transverse distribution is considered zero emittance at the beginning as shown in \begin_inset LatexCommand \ref{cap:Ch6_2Emittance Compensation} \end_inset (a). As the beam drifts from position (a) in figure \begin_inset LatexCommand \ref{cap:Ch6_2Emittance Compensation} \end_inset , the \begin_inset Formula $\zeta$ \end_inset dependent radial space charge forces cause the projected phase space to increase, with the radius of the core increasing more than the head or tail of the bunch. This space charge dominated expansion of the electron phase space is shown in \begin_inset LatexCommand \ref{cap:Ch6_2Emittance Compensation} \end_inset (b). As this beam travels through a thin solenoid focusing field, the individual slices of the electron beam ( see figure \begin_inset LatexCommand \ref{cap:Ch6_1Projected-Phase-Space} \end_inset ) receive different focusing kick from the solenoid field. In an individual slice, this kick is proportional to the electron's x position. This is shown in figure \begin_inset LatexCommand \ref{cap:Ch6_2Emittance Compensation} \end_inset (c). The electron beam is now convergent, and the space charge forces defocus it again as it moves through drift space after the lens. The phase space trajectory followed by these slices is shown in \begin_inset LatexCommand \ref{cap:Ch6_2Emittance Compensation} \end_inset (d). At some longitudinal distance z \begin_inset Formula $_{\textrm{o}}$ \end_inset , in the drift region, the head, tail and the core realign into a zero emittance beam. In practice it will be a low emittance beam, with the emittance being determine d by non linear effects in the gun and the initial thermal emittance at the cathode. \layout Standard The analysis presented in the above sections work under the approximation of the linear regime (we neglected the second term on right hand side of equation \begin_inset LatexCommand \ref{eq:Ch6_35EnvelopeFinal} \end_inset ). In real accelerators there are additional effects caused by non uniformity of the beam, hot spots, non linear focusing terms in the RF gun and solenoids. As analytical models can not incorporate all these effects we resort to experiments and simulations for a more comprehensive understanding of electron beam evolution. \layout Section \paragraph_spacing double Experiments \begin_inset LatexCommand \label{sec:ExperimentsEmittance} \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Experimental_SchemeZhou.eps display none clip \end_inset \layout Caption Schematic layout for experimental emittance measurements. \begin_inset LatexCommand \label{cap:Ch6_F3EmittanceSchematic} \end_inset \layout Standard BPM is the beam profile monitor and Q1 through Q6 denote six Quadrapoles. The quadrapole Q6 is scanned to measure transverse emittance \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Charge_vs_LaserEnergy.eps display none clip \end_inset \layout Caption Charge vs. laser energy. \begin_inset LatexCommand \label{cap:Ch6_F4Undistorted-laser-beam} \end_inset \layout Standard Varying the laser energy has a linear effect on the variation of charge in the electron beam. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Subsection Emittance Measurement Technique \begin_inset LatexCommand \label{sub:Emittance-measurement-technique} \end_inset \layout Standard The transverse emittance is measured at a beam profile monitor (BPM) by scanning quadrupole strength. A schematic sketch is shown in the figure \begin_inset LatexCommand \ref{cap:Ch6_F3EmittanceSchematic} \end_inset . The BPM comprises a phosphorescent screen, and a charged couple device (CCD) camera with a frame grabber to record data. The detailed description of the measurement technique at the ATF is given in \begin_inset LatexCommand \ref{sub:Measurement-of-Emittance} \end_inset . \layout Standard \paragraph_spacing double We start by measuring the emittance of the electron beam when the cathode surface is illuminated by undistorted laser beam. This gives a reference point, and we can compare the effect of non-uniformity on the electron beam emittance. Figure \begin_inset LatexCommand \ref{cap:Ch6_F4Undistorted-laser-beam} \end_inset (a) shows an image of the laser beam on the cathode and the \begin_inset LatexCommand \ref{cap:Ch6_F4Undistorted-laser-beam} \end_inset (b) plots of the charge vs. the laser energy. The latter indicates the linear relationship between the laser energy and the charge on the electron beam. \layout Subsection Imposed Non Uniformity \begin_inset LatexCommand \label{sub:Imposed-non-uniformity} \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/OnCathode_VariousLinesC.eps display none clip \end_inset \layout Caption Image of the cathode illuminated by various non uniform distributions \begin_inset LatexCommand \label{cap:Ch6_F5ImagesNonUniform} \end_inset \layout Standard Figure shown in (a) on the DMD gives 4 lines on cathode surface. (b) and (c) create 8 and 12 lines respectively \end_inset \layout Standard \paragraph_spacing double To study the effect of laser distortion, we impose upon the laser distribution a controlled modulation, using the DMD as a spatial light modulator. This modulation is created by sending to the digital mirror device, an appropriate image. These digital images are generated by matlab scripts and we vary the number of lines in these images. The bit map image can then be read into the DMD, using the client application (ref: \begin_inset LatexCommand \ref{sec:Ch5_SoftwareDMD} \end_inset ). This application converts the images into a format acceptable by the DMD. These images, seen in \begin_inset LatexCommand \ref{cap:Ch6_F5ImagesNonUniform} \end_inset (a), (b) and (c) distort the laser beam, as can be seen in (d), (e) an (f) sub-figures of the figure \begin_inset LatexCommand \ref{cap:Ch6_F5ImagesNonUniform} \end_inset . The number of lines created on the cathode surface is limited by the optics and the focusing systems. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/ExperimentalSetup1.eps display none clip \end_inset \layout Caption DMD device seen in the optical beam line \begin_inset LatexCommand \label{cap:Ch6_F6DMDInBeamline} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double For the purpose of our investigation, we use three such representative images, which produce 4, 8 and 12 lines on the cathode surface within a 2.4 \begin_inset Formula $mm$ \end_inset diameter. For the rest of the analysis, we name the distributions, of the laser and the electron beam by a line number. For example a distribution as seen in figure \begin_inset LatexCommand \ref{cap:Ch6_F5ImagesNonUniform} \end_inset (d) is called 4 lines and the distribution corresponding to figure \begin_inset LatexCommand \ref{cap:Ch6_F5ImagesNonUniform} \end_inset (e) 8 lines respectively. These intensity distributions are referred to as various modulations. As can be seen from figure \begin_inset LatexCommand \ref{cap:Ch6_F6DMDInBeamline} \end_inset the experimental set up places the DMD at an 45 degree angle, hence we create images tilted at the same angle to obtain straight line distributions on the cathode surface. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Ch6_EmittancevsLines.eps display none clip \end_inset \layout Caption Emittance decreases as the number of lines increases \begin_inset LatexCommand \label{cap:Ch6_F7EmittancevsLines} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard Figure \begin_inset LatexCommand \ref{cap:Ch6_F7EmittancevsLines} \end_inset shows the result of this experiment. The plot shows the measured emittance of the electron beam vs the various distributions corresponding to 4, 8 and 12 lines on the cathode. The results are tabulated in table \begin_inset LatexCommand \ref{cap:TableEXPEmittance vsLines} \end_inset . \begin_inset Float table wide false collapsed false \layout Caption Measured Emittance decreases as number of line increases \begin_inset LatexCommand \label{cap:TableEXPEmittance vsLines} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard No of Lines \end_inset \begin_inset Text \layout Standard Emittance [mm-mrad] \begin_inset Formula $\epsilon_{x}$ \end_inset \end_inset \begin_inset Text \layout Standard Emittance[mm-mrad] \begin_inset Formula $\epsilon_{y}$ \end_inset \end_inset \begin_inset Text \layout Standard Undistorted beam \end_inset \begin_inset Text \layout Standard 0.97 \end_inset \begin_inset Text \layout Standard 1.32 \end_inset \begin_inset Text \layout Standard 4 \end_inset \begin_inset Text \layout Standard 1.5 \end_inset \begin_inset Text \layout Standard 2.6 \end_inset \begin_inset Text \layout Standard 8 \end_inset \begin_inset Text \layout Standard 0.8 \end_inset \begin_inset Text \layout Standard 1.3 \end_inset \begin_inset Text \layout Standard 12 \end_inset \begin_inset Text \layout Standard 0.87 \end_inset \begin_inset Text \layout Standard 1.38 \end_inset \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Subsection Varying Contrast \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Experimental_LaserEnergy_vsCharge.eps display none clip \end_inset \layout Caption Charge vs. laser energy. \begin_inset LatexCommand \label{cap:Ch6_F7Charge-vs.-Laser} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard Varying the intensity of the laser beam effectively controls the charge of the electron beam. This variation for an undistorted laser beam can be seen in figure \begin_inset LatexCommand \ref{cap:Ch6_F7Charge-vs.-Laser} \end_inset . For the non-uniform laser beam distributions we vary the contrast by creating images with appropriate contrast in the intensity of the lines. \layout Standard We vary the contrast of the impinging images, by changing the ratio between the dark and bright lines. Thus a 50% implies that the white lines correspond to one and the dark line to 0.5 (i.e allows 50% less light compared to be incident on the cathode surface compared to the bright strip) on a created image. Such a bitmap is then dithered \begin_inset Foot collapsed false \layout Standard dither is a technique to create indexed image ( an array of M X N that stores the image and the color map being stored separately) from RGB format (an array of MxNx3) that stores red , green and blue for each pixel. \end_inset to create a modulated image as shown in figure \begin_inset LatexCommand \ref{cap:Ch6_F9Created-modulation} \end_inset . We measure the emittance for changing contrast. One such representative plot is shown in figure \begin_inset LatexCommand \ref{cap:Ch6_F10Emittance-vs.-modulation} \end_inset . \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/ModulationExample.eps display none clip \end_inset \layout Caption Created contrast of intensity \begin_inset LatexCommand \label{cap:Ch6_F9Created-modulation} \end_inset \layout Standard (a) and (b) show the intensity contrast created on indexed images (c) and (d) show the corresponding uniformity distortion as visible on the cathode surface. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Ch6_EmittancevsMod8Lines.eps display none clip \end_inset \layout Caption Emittance vs. contrast. \begin_inset LatexCommand \label{cap:Ch6_F10Emittance-vs.-modulation} \end_inset \layout Standard The emittance of electron-beam decreases as the contrast is changed on the incident image. This is a representative plot collected by measuring the emittance at 25%, 50%, 75%, and 100% contrast between the dark and white lines of a image with 8 line distribution. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float table wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Caption Emittance vs. Contrast \begin_inset LatexCommand \label{cap:Emittance-vs.-Modulation} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard contrast \end_inset \begin_inset Text \layout Standard Emittance [mm-mrad] \begin_inset Formula $\epsilon_{x}$ \end_inset \end_inset \begin_inset Text \layout Standard Emittance[mm-mrad] \begin_inset Formula $\epsilon_{y}$ \end_inset \end_inset \begin_inset Text \layout Standard 25% \end_inset \begin_inset Text \layout Standard 1.75 \end_inset \begin_inset Text \layout Standard 1.89 \end_inset \begin_inset Text \layout Standard 50% \end_inset \begin_inset Text \layout Standard 1.0 \end_inset \begin_inset Text \layout Standard 1.6 \end_inset \begin_inset Text \layout Standard 75% \end_inset \begin_inset Text \layout Standard 0.8 \end_inset \begin_inset Text \layout Standard 1.5 \end_inset \begin_inset Text \layout Standard 100% \end_inset \begin_inset Text \layout Standard 0.8 \end_inset \begin_inset Text \layout Standard 1.3 \end_inset \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Subsection Summary of Experimental Results \layout Standard \paragraph_spacing double We find, as expected from previous experimental studies, that the introduction of a macroscopic non-uniformity in the laser beam affects the electron beam emittance.An e-beam created by a non uniform laser beam has higher emittance as compared to a uniform laser beam. Our experiments consistently showed that adding fine grained structure to the macroscopic distortions does not lead to further increase of emittance. On the contrary, we find that, as we reduce the size of the macroscopic non-uniformity the emittance begins to improve. We investigate the effect further by simulating similar distributions in an electron beam. \layout Section Parmela Simulations \begin_inset LatexCommand \label{sec:Parmela-Simulations} \end_inset \layout Standard \paragraph_spacing double PARMELA, an acronym for the phrase, \begin_inset Quotes eld \end_inset Phase and Radial Motion in Electron Linear Accelerators \begin_inset Quotes erd \end_inset , is a multi-particle tracking code that is used for electron-linac beam simulations. It simulates beam dynamics of beam transport and accelerator systems by integrating trajectories of macro-particles through the fields using phase (time) as the independent variable. For the field distributions PARMELA requires input files generated by other dedicated codes like SUPERFISH \begin_inset Foot collapsed false \layout Standard SUPERFISH generates RF fields \end_inset and POISSON \begin_inset Foot collapsed false \layout Standard POISSON is used for static magnetic fields \end_inset , which solve finite difference equations for the Maxwell equations, with appropriate material properties and boundary conditions. \layout Standard \paragraph_spacing double Parmela utilizes a user defined input file, \emph on input.acc \begin_inset Foot collapsed false \layout Standard The .acc extension ensures that PARMELA recognises it as a PARMELA exe. \layout Standard All other file names listed in italics this section are defaults and can be changed by accessing the Initialisation file with extension .INI \end_inset , \emph default to model the beam line that is to be simulated. At the end of each space charge impulse and/ or beam line element, Parmela writes to an ASCII text file, \emph on outpar.txt. \emph default The graphics program PARGRAF is used to view the particle distribution, in phase space at the end of each element of the beam-line. The control file, \emph on simple \emph default . \emph on pgf \emph default , is user defined to determine elements of the beam line, where output is needed. PARGRAF outputs an ASCII text file, \emph on outgraf, \emph default which contains the beam emittance at the end of each requested element. \layout Standard \paragraph_spacing double PARMELA produces transverse and longitudinal space charge impulses to the macro-particles of the electron beam at phase steps determined by \emph on input.acc. \emph default The space charge impulses are calculated in the rest frame of the reference particle. A relativistic boost is then given to transform the electrostatic field of the electron bunch from the rest frame of the reference particle to the laboratory frame. The space charge impulses are computed by the point to point method, where in Coulomb's law is used to calculate the space charge impulse for each individual macro-particle due to all other macro-particles. \layout Standard \paragraph_spacing double We model various types of transverse laser profiles, keeping the longitudinal laser profile a Gaussian. One or more INPUT lines can specify the initial distribution for PARMELA. To model the undistorted laser beam, we use the input with standard card type 9, which provides Gaussian or uniform distribution with the same width in both transverse directions, and a different Gaussian or uniform distribution in phase. To model the transverse distributions capturing various kinds of non-uniformity , the type 40 INPUT card was used, which allows PARMELA to read in particle coordinate data from a specified file. We generate the specified file, independently, using matlab scripts which capture the various kinds of imposed non-uniformity. \layout Standard The total number of macro-particles used to model the electron beam was 10,000. \layout Subsection Simulation Results \layout Standard \paragraph_spacing double As in experiments we start by simulating the emittance evolution of the electron beam, when the cathode is irradiated by an undistorted laser beam. For this the electrons are emitted randomly with a profile that is Gaussian in time and radius, as for the undistorted laser pulse. The distribution of the electrons in the transverse and longitudinal directions is shown in figure \begin_inset LatexCommand \ref{cap:Ch6_F10ParmelaPhaseSpace} \end_inset . Figure \begin_inset LatexCommand \ref{cap:Ch6_F11ParmelaTraceSpace} \end_inset shows the electron beam size in configuration space of (x-y). As the e-beam moves downstream from the gun, the beam size increases, goes through a maximum and then begins to decrease as it gets focused due to the effect of a solenoid in the system and reaches a minimum value. At this point it is usually injected into the accelerator. In the accelerator the speeds are ultra-relativistic and the space charge effects become negligible, thereby freezing the emittance to this minimum value. The evolution of emittance of an electron beam corresponding to the undistorted laser beam is shown in \begin_inset LatexCommand \ref{cap:Ch6_F12EmittanceNoMod} \end_inset . \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash centering \layout Standard \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/PhaseSpace.eps display none keepAspectRatio BoundingBox 150bp 0bp 500bp 337bp clip \end_inset \layout Caption Electron beam profile space as simulated in PARMELA \begin_inset LatexCommand \label{cap:Ch6_F10ParmelaPhaseSpace} \end_inset \layout Standard (a) and (b) shows the \begin_inset Formula $x'-x$ \end_inset and \begin_inset Formula $y'-y$ \end_inset transverse trace space projections, respectively of the six dimensional phase space. (c) shows the longitudinal distribution in the co-ordinates of energy differenc e \begin_inset Formula $(e-e_{s})$ \end_inset vs phase difference \begin_inset Formula $(\phi-\phi_{s}$ \end_inset ) of the beam, where \begin_inset Formula $e_{s}$ \end_inset is the energy of the synchronous electron and \begin_inset Formula $\phi_{s}$ \end_inset is the phase of the synchronous electron. \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/electron_beam_evolution.eps display none clip \end_inset \layout Caption Electron beam evolution for the ATF-Photo-injector simulated using PARMELA. \begin_inset LatexCommand \label{cap:Ch6_F11ParmelaTraceSpace} \end_inset \layout Standard (a) shows the transverse phase space of the electron beam as it is ejected at the cathode. (b) shows the increase in the transverse beam size. (c) The beam size continues to increase and passes through a maximum as shown in (c). (d) shows the effect of the solenoid, and the transverse beam size begins to decrease. (e) The transverse beam size goes through a minimum. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Emittance_NoModulation.eps display none clip \end_inset \layout Caption Emittance of electron beam corresponding to undistorted laser beam. \begin_inset LatexCommand \label{cap:Ch6_F12EmittanceNoMod} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double To compare simulations with results from experiments, we generate particle distributions in accordance with the various non-uniformities of the experiment s. In accordance with the nomenclature followed in experimental set up, we call the various distributions 4 lines, 8 lines and 12 lines, which are representative of the lines created on the cathode-surface. Such a distribution of 4lines, generated by using matlab scripts, is shown in figure \begin_inset LatexCommand \ref{cap:Ch6F13_4-line-particle} \end_inset . This distribution serves as the transverse distribution (x-y) of the electron beam. The distribution along the longitudinal dimension is a Gaussian as shown in figure \begin_inset LatexCommand \ref{cap:Ch6F13_4-line-particle} \end_inset (b). Such a created distribution is read into the beam diagnostic code PARMELA, and the electron beam evolution is followed. The emittance of the beam as it evolves through the various elements is plotted and is showed in figure \begin_inset LatexCommand \ref{cap:Ch6_F12EmittanceNoMod} \end_inset . The emittance reaches a maximum as the electron beam evolves and then begins to decrease due to compensation by the solenoid. It goes through a minimum, before it begins to increase again. This minimum is the least emittance of the evolving e-beam and will be referred to as the emittance of the distribution. We then repeat the simulations for particle distributions, in the transverse direction, having 8 and 12 lines, figure \begin_inset LatexCommand \ref{cap:Ch6_F17Various-distributions} \end_inset shows the distributions generated. Figure \begin_inset LatexCommand \ref{cap:Ch6_F18EmittanceEvolveVariousDist} \end_inset shows the emittance of all these distributions plotted together. This plot also includes the plot of emittance of the undistorted beam. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/4lineAndHorizontal.eps display none clip \end_inset \layout Caption 4 line particle distribution generated using matlab \begin_inset LatexCommand \label{cap:Ch6F13_4-line-particle} \end_inset \layout Standard (a) shows a 4line distribution within a circular diameter of 0.24 cm, which was created corresponding to electron beam diameter of 2.4 mm. (b) shows the horizontal distribution of the particles.The longitudinal distribution is a Gaussian with a variance of 2.4 mm , which corresponds to 3.4 degrees. \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/electron_beam_evolution4Lines.eps display none clip \end_inset \layout Caption Evolution of electron beam in configuration space with a non-uniform distributio n for the ATF photo-injector simulated in PARMELA \begin_inset LatexCommand \label{cap:Ch6F15_Evolution4Lines} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/4lines_emittance.eps display none clip \end_inset \layout Caption Simulated emittance vs. distance for the 4 line distribution \begin_inset LatexCommand \label{cap:Ch6_F16Emittancevsdistance4LINE} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/VariousLinesForParmela.eps display none clip \end_inset \layout Caption Various distributions used to simulate corresponding experimental distributions in Parmela \begin_inset LatexCommand \label{cap:Ch6_F17Various-distributions} \end_inset \layout Standard (a) shows a 4 line distribution, (b) shows a 8 line distribution and (c) shows a 12 line distribution imposed on a circular beam of 2.4 mm diameter. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/DistVariousLines.eps display none scale 75 keepAspectRatio clip \end_inset \layout Caption Plot of emittance for various distributions \begin_inset LatexCommand \label{cap:Ch6_F18EmittanceEvolveVariousDist} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard Once again the minimum emittance is extracted and we make a table comparing flat beam to distorted beam. \layout Standard \begin_inset Float table wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Caption Emittance vs number of lines from simulations \begin_inset LatexCommand \label{cap:Ch6_EmittanceVsLines} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard Number Of Lines \end_inset \begin_inset Text \layout Standard Emittance [cm-mrad] \begin_inset Formula $\epsilon_{x}$ \end_inset \end_inset \begin_inset Text \layout Standard Emittance[cm-mrad] \begin_inset Formula $\epsilon_{y}$ \end_inset \end_inset \begin_inset Text \layout Standard 4lines \end_inset \begin_inset Text \layout Standard 1.17 \end_inset \begin_inset Text \layout Standard 2.3 \end_inset \begin_inset Text \layout Standard 8 lines \end_inset \begin_inset Text \layout Standard 1.8 \end_inset \begin_inset Text \layout Standard 1.07 \end_inset \begin_inset Text \layout Standard 12 lines \end_inset \begin_inset Text \layout Standard 1.8 \end_inset \begin_inset Text \layout Standard 1.02 \end_inset \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Subsection Varying Contrast \layout Standard We simulate the modulated distributions, to verify the experimental results. The modulated particle distributions are created by matlab scripts. Figure \begin_inset LatexCommand \ref{cap: Ch6_F19MatlabModulated} \end_inset shows such particle distributions. Figure \begin_inset LatexCommand \ref{cap: Ch6_F19MatlabModulated} \end_inset (a) shows a 8 line distribution, with 50% contrast. This means that the number of particles, between the strips in a, have 50% particles compared adjacent strips, figure \begin_inset LatexCommand \ref{cap: Ch6_F19MatlabModulated} \end_inset (b) shows the same 8 line distribution with 75% contrast. In figure \begin_inset LatexCommand \ref{cap:Ch6_F208Lines_em_vs_mod} \end_inset , we plot the evolution of the emittance vs. the distance for various contrasts of a 8 line distribution. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/VariousMod8Lines.eps display none clip \end_inset \layout Caption Contrasts generated in matlab \begin_inset LatexCommand \label{cap: Ch6_F19MatlabModulated} \end_inset \layout Standard Particle distributions which mimic the experimental distributions were generated in matlab. The distributions show 25%, 50%, 75% and 100% contrast in a 8 line distribution with 1.2 mm radius and \begin_inset Formula $3.4^{o}$ \end_inset variance in the phase distribution. Only the x,y distribution is shown here. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/emittance_x_vs_mod8lines.eps clip \end_inset \layout Caption Emittance for various contrasts for a 8 line distribution. \begin_inset LatexCommand \label{cap:Ch6_F208Lines_em_vs_mod} \end_inset \layout Standard This plot shows the evolution of emittance for various contrasts in a 8 line distribution. On the y axis is the plot of emittance in the x direction and on the x axis is distance in cm. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section Comparison of Simulations and Experiments \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/CompareExpAndSims.eps display none scale 75 clip \end_inset \layout Caption Comparison of simulated vs. experimental results. \begin_inset LatexCommand \label{cap:Ch6_F21compare} \end_inset \layout Standard The plot shows the emittance vs the number of lines. The red line is the experimental data and the blue line is the simulated numbers. The emittance plotted here is the geometric mean of the emittance in the x and y directions. The experimental accuracy is about 15%. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard We find that simulations and experimental results are in reasonable agreement. It is verified that the presence of macroscopic non uniformities in the laser beam distribution affects the emittance of the electron beam. (The emittance of the electron beam with 4 line distribution emittance is larger than the emittance of the electron beam corresponding to the undisturbed laser beam). We further establish that, fine graining of these non-uniformities diminishes the emittance growth. Hence the emittance corresponding to the 8 line distribution improves compared to the 12 lines distribution. Experimental results even so performed at the state of the art facility, have significant relative errors in emittance measurement. Further the simulations were performed in accordance to the best known parameters while real emittance was measured at the ATF accelerator which was lightly optimized in years of operation. Nevertheless, the simulations verify behaviours very similar to that observed in the experiments. \layout Standard Increase of depth of modulation for a given structure and space charge increases beam emittance. It is also clear from simulations as well as experiments that variations in density at a scale much smaller than the beam size ( \begin_inset Formula $\frac{1}{10}$ \end_inset or smaller) do not increase beam emittance significantly. On the other hand macroscopic features (4 line distribution, for example) worsens the emittance by a factor of two. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Compare8linesModulated.eps display none clip \end_inset \layout Caption Comparison of emittance in simulated and experimental data for a 8 line distribution across various contrasts. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section Understanding of Results \begin_inset LatexCommand \label{sec:PhysicsOfResults} \end_inset \layout Standard We propose a simple model to gain physical insight. An intensity modulation in the distribution of the drive laser beam causes a modulation in the density distribution of the electron beam. The e-beam density distribution mimics the intensity modulation of the laser beam. We model density distribution as an oscillatory function in one dimension in accordance with the created intensity distributions, that were investigated experimentally and numerically. Thus the electron beam density may be written as \layout Standard \begin_inset Formula \begin{equation} \rho=\rho_{o}(1+a\cos(k_{m}x)),\label{eq:Ch6_DensityDist}\end{equation} \end_inset \layout Standard where \begin_inset Formula $a$ \end_inset is the amplitude of the modulation and \begin_inset Formula $k_{m}=\frac{2\pi}{\lambda_{m}}$ \end_inset is the wave vector associated with the modulation. \layout Standard In the rest frame of the electron the Laplace equation in one dimension is \begin_inset Formula \begin{equation} \nabla^{2}\psi=\frac{\partial^{2}\psi}{\partial x^{2}}=4\pi(\rho_{o}+\rho_{mod}).\label{eq:Ch6_Laplace}\end{equation} \end_inset \layout Standard where \begin_inset Formula $\rho_{o}$ \end_inset is the constant term that accounts for the linear forces and \begin_inset Formula $\rho_{mod}$ \end_inset is the contribution due to non linear effects of the one dimensional modulation. The potential thus may be written as \begin_inset Formula \begin{equation} \psi=4\pi\rho_{o}\frac{x^{2}}{2}-\frac{4\pi}{k_{m}^{2}}\rho_{o}a\cos(k_{m}x)+C\label{eq:Ch6Potential}\end{equation} \end_inset where, for an electron beam of radius \begin_inset Formula $r_{b}$ \end_inset and bunch length \begin_inset Formula $l_{b}$ \end_inset the charge density \begin_inset Formula $\rho_{o}=\frac{N_{e}e}{\pi r^{2}\gamma l_{b}}$ \end_inset . \layout Standard We may ignore the first term on right hand side of equation \begin_inset LatexCommand \ref{eq:Ch6Potential} \end_inset , which corresponds to linear forces as we wish to study the effects of the modulation. Writing the contribution of the modulation to the potential as \begin_inset Formula $\psi_{mod}$ \end_inset , we get \layout Standard \begin_inset Formula \begin{equation} \tilde{\psi}_{mod}=\frac{1}{k_{m}^{2}}4\pi\rho_{o}(1+a\cos(kx)).\label{eq:Ch6_PotentialMod}\end{equation} \end_inset \layout Standard Substituting for \begin_inset Formula $\rho_{o}$ \end_inset and \begin_inset Formula $k_{mod}$ \end_inset in equation \begin_inset LatexCommand \ref{eq:Ch6_PotentialMod} \end_inset we get \begin_inset Formula \begin{equation} \tilde{\psi}_{mod}=\frac{N_{e}e}{\pi r_{b}^{2}\gamma l_{b}}\frac{\lambda_{m}^{2}}{\pi^{2}}(1+a\cos(kx)).\label{eq:Ch6_PotentialMod2}\end{equation} \end_inset \layout Standard On the time scale corresponding to one plasma oscillation the electron beam experiences emittance growth due to this energy contribution. This energy contribution imparts transverse momentum to the electrons, which in turn leads to emittance growth. This may be estimated as \layout Standard \begin_inset Formula \begin{equation} \frac{\Delta p_{\perp}^{2}}{2m}=\Delta E_{mod}=e\tilde{\psi}_{mod}.\label{eq:Ch6_TransverseMomentum}\end{equation} \end_inset \layout Standard where \begin_inset Formula $\Delta p_{\perp}$ \end_inset is the transverse momentum of an electron and \begin_inset Formula $\Delta E_{mod}$ \end_inset is the growth in energy due to this transverse momentum. \layout Standard Substituting \begin_inset Formula $\tilde{\psi}_{mod}$ \end_inset from equation \begin_inset LatexCommand \ref{eq:Ch6_PotentialMod2} \end_inset we get \begin_inset Formula \begin{equation} \frac{\Delta p_{\perp}^{2}}{2m}=\Delta E_{mod}=\frac{N_{e}e^{2}}{\pi^{2}r_{b}^{2}\gamma l_{b}}\frac{\lambda_{m}^{2}}{\pi^{2}}(1+a\cos(kx)).\label{eq:Ch6_TransverseMomentum2}\end{equation} \end_inset \layout Standard Rearranging terms in equation \begin_inset LatexCommand \ref{eq:Ch6_TransverseMomentum2} \end_inset we write \begin_inset Formula \begin{equation} \frac{\Delta p_{\perp}^{2}}{2m}=\frac{mc^{2}}{mc^{2}}\frac{2N_{e}e}{\pi r_{b}^{2}\gamma l_{b}}\frac{^{2}\lambda_{m}^{2}}{\pi^{2}}a(\cos^{2}(kx/2)).\label{eq:Ch6_TransverseMometum3}\end{equation} \end_inset Substituting \begin_inset Formula $\frac{e^{2}}{mc^{2}}$ \end_inset as the classical radius of an electron \begin_inset Formula $r_{e}$ \end_inset , \begin_inset Formula \[ \frac{\Delta p_{\perp}^{2}}{2m}=2mc^{2}r_{e}\frac{N_{e}}{\pi^{2}}\frac{\lambda_{m}^{2}}{r_{b}^{2}\gamma l_{b}}a\cos^{2}(kx/2).\] \end_inset \layout Standard Recapitulating \begin_inset Formula $x'=\frac{\Delta p_{\perp}}{mc}$ \end_inset , we write \begin_inset Formula \begin{equation} x'=\frac{\Delta p_{\perp}}{mc}=\frac{2\lambda_{m}}{\pi r_{b}}\sqrt{\frac{N_{e}r_{e}a}{\gamma l_{b}}}\cos(kx/2).\label{eq:Ch6_div}\end{equation} \end_inset \layout Standard Emittance is defined as \begin_inset Formula \begin{equation} \epsilon^{2}=\langle x^{2}\rangle\langle x'^{2}\rangle.\label{eq:Ch6_em}\end{equation} \end_inset Substituting for \begin_inset Formula $\langle x^{2}\rangle=\frac{r_{b}^{2}}{4}$ \end_inset and \begin_inset Formula $\langle x'^{2}\rangle=2\frac{\lambda_{m}^{2}}{\pi^{2}r_{b}^{2}}\frac{N_{e}r_{e}a}{\gamma l_{b}}$ \end_inset in equation \begin_inset LatexCommand \ref{eq:Ch6_em} \end_inset , we obtain \begin_inset Formula \begin{equation} \Delta\epsilon=\frac{\lambda_{m}}{\pi}\sqrt{\frac{1}{2}\frac{N_{e}r_{e}a}{\gamma l_{b}}}\label{eq:Ch6_em2}\end{equation} \end_inset \layout Standard We test our model by substituting numbers from the ATF experimental parameters of the electron beam. The radius of the beam \begin_inset Formula $r_{b}$ \end_inset is 1.2 mm, the bunch length is \begin_inset Formula $2.4$ \end_inset mm and the energy of the electron beam is \begin_inset Formula $4.5$ \end_inset MeV (implies \begin_inset Formula $\gamma=10.4$ \end_inset ) and for a typical charge of 160 pC, we estimate the contribution to the emittance due to the modulation as given in table \begin_inset LatexCommand \ref{cap:Ch6_Estimated-And-ObservedTable} \end_inset . \begin_inset Float table wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Caption Estimated And Observed Emittance. \begin_inset LatexCommand \label{cap:Ch6_Estimated-And-ObservedTable} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard Number Of Lines \end_inset \begin_inset Text \layout Standard \begin_inset Formula $\Delta\epsilon_{mod}$ \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula $\epsilon_{est}=\sqrt{\Delta\epsilon_{mod}^{2}+\epsilon_{in}^{2}}$ \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula $\epsilon_{observed}$ \end_inset in mm-mrad \end_inset \begin_inset Text \layout Standard 4 \end_inset \begin_inset Text \layout Standard 1.43 \end_inset \begin_inset Text \layout Standard 1.75 \end_inset \begin_inset Text \layout Standard \begin_inset Formula $1.9\pm0.27$ \end_inset \end_inset \begin_inset Text \layout Standard 8 \end_inset \begin_inset Text \layout Standard 0.71 \end_inset \begin_inset Text \layout Standard 1.23 \end_inset \begin_inset Text \layout Standard \begin_inset Formula $1\pm0.2$ \end_inset \end_inset \begin_inset Text \layout Standard 12 \end_inset \begin_inset Text \layout Standard 0.36 \end_inset \begin_inset Text \layout Standard 1.16 \end_inset \begin_inset Text \layout Standard \begin_inset Formula $1.2\pm0.3$ \end_inset \end_inset \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard We find that the estimated emittance is in good agreement with the observed emittance. \layout Chapter Photon Beams \layout Section \paragraph_spacing double Introduction \layout Standard \paragraph_spacing double For the second part of this thesis we focus on photon beams. Further more we restrict ourselves to beams of coherent radiation generated by free electron lasers. Free electron lasers are devices that generate tunable high power radiation spanning wavelengths from millimeters to ultraviolet and potentially X-rays. We start the discussion with the principle of a free electron laser in \begin_inset LatexCommand \ref{sec:Ch7_FEL-Theory} \end_inset and include concepts related to basic principles, spontaneous and stimulated emissions in context of free electron lasers. In the subsequent section \begin_inset LatexCommand \ref{Ch7_OK} \end_inset we discuss a modification over the conventional free electron laser. This modified free electron laser is called an optical klystron. This special mention and detailed discussion for an optical klystron is warranted by the fact that all operational storage ring free electron lasers utilize this set up, including the Duke storage ring free electron laser. Section \begin_inset LatexCommand \ref{sec:Ch7Duke-Storage-Ring} \end_inset carries an overview of the OK-4/Duke storage ring free electron laser, henceforth referred to as OK-4/Duke SRFEL. The subsequent sections report on experimental and numerical techniques implemented at the OK-4/Duke SRFEL. We start with section \begin_inset LatexCommand \ref{sec:Ch7_Giant-Pulses-at} \end_inset which introduces the idea of redistribution of FEL peak power at the expense of average power. This mode of operation, achieved by gain modulation techniques, is called giant pulse mode. We report on simulations as well as experimental results using gain modulation to produce giant pulses. In section \begin_inset LatexCommand \ref{sec:Ch7_Harmonics} \end_inset , we extend the discussion to harmonic generation in storage ring free electron lasers and present data from proof of principle experiments carried out at OK-4/Duke SRFEL. \layout Section \paragraph_spacing double FEL Theory \begin_inset LatexCommand \label{sec:Ch7_FEL-Theory} \end_inset \layout Subsection \paragraph_spacing double Basic Principle \layout Standard \paragraph_spacing double A free electron laser uses a beam of free \begin_inset Foot collapsed false \layout Standard The electrons are not bound to atoms or molecules,hence the term free electron laser \end_inset relativistic electrons to generate very high intensity electromagnetic radiation.The basic components of a traditional free electron laser are:wiggler magnet (a series of poles with alternating polarity), the electron beam (produced by either a storage ring or an accelerator) and a resonator cavity. Figure \begin_inset LatexCommand \ref{FELChapter_FElSchematic} \end_inset shows the basic outline of a free electron laser. The wiggler magnet produces a transverse magnetic field which is static in time but varies sinusoidally in space. The amplitude of this magnetic field \begin_inset Formula $(B_{w})$ \end_inset is typically a few kilogauss, and the period \begin_inset Formula $(\lambda_{w})$ \end_inset is typically a few cm. The propagating e-beam oscillates transversely due to the the magnetic field of the wiggler, and emits spontaneous radiation. The transversely oscillating electron beam then interacts with the co-propagati ng optical wave and can be made to transfer its kinetic energy to the optical field. To run as an oscillator, the FEL needs the resonator cavity which provides positive feedback. The cavity usually has mirrors at both ends. Radiation is typically coupled-out by putting a partially reflecting mirror or in some cases through a hole in the mirror. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel1.eps clip \end_inset \layout Caption Schematic of a free-electron laser(FEL) \begin_inset LatexCommand \label{FELChapter_FElSchematic} \end_inset \layout Standard The figure shows the wiggler magnet that produces a static but spatially sinusoidal magnetic field, and the mirrors that provide a positive feedback. A relativistic electron beam is injected into the wiggler magnet, which consists of a row of magnets with alternating polarity. The synchrotron light, induced by the wiggling of the electrons, is captured by cavity mirrors and coherently amplified to produce lasing. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Subsection \paragraph_spacing double Spontaneous Emission \begin_inset LatexCommand \label{sub:Ch7_Spontaneous-emission} \end_inset \layout Standard \paragraph_spacing double The core of the FEL is the wiggler which produces a magnetic field static in time but varying in space. The magnetic field of the wiggler is perpendicular to the direction of the electron beam. The wiggler magnets are so arranged that they change polarity a number of times along the wiggler length. The magnetic field from this arrangement of magnets varies sinusoidally in space and can be represented by \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \vec{B}_{w}=B_{o}\cos(k_{w}z)\hat{y}\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $B_{o}$ \end_inset is the peak of the magnetic field and \begin_inset Formula $k_{w}=\frac{2\pi}{\lambda_{w}}$ \end_inset and \begin_inset Formula $\lambda_{w}$ \end_inset is the length of wiggler period. An electron injected at the end of the wiggler and travelling down the length of the wiggler (say, along z axis) experiences a Lorentz force due to this transverse magnetic field. As this magnetic field is oscillating in space the electron also begins to oscillate in the x-z plane. Since any oscillating electric charge emits radiation, the electrons in a wiggler emit radiation. The wavelength of this radiation can be calculated by considering the electron in its rest frame. In this reference frame, the electron is at rest and the wiggler moves toward it with a velocity \begin_inset Formula $v_{z}\approx c$ \end_inset , where \begin_inset Formula $v_{z}$ \end_inset is the z component of the mean electron velocity. The wiggler period in this frame is \begin_inset Formula $\lambda_{w}/\gamma_{z}$ \end_inset , where \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \gamma_{z}=\frac{1}{\sqrt{1-\beta_{z}^{2}}}\end{equation} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \beta_{z}=\frac{v_{z}}{c}\end{equation} \end_inset \layout Standard \paragraph_spacing double The magnetic field of the wiggler appears to the electron as an electromagnetic field with wavelength \begin_inset Formula $\frac{\lambda_{w}}{\gamma_{z}}$ \end_inset . The electron classically radiates at the same wavelength. In the laboratory frame of references the radiation is Doppler up shifted by another factor of \begin_inset Formula $2\gamma_{z}$ \end_inset (Appendix \begin_inset LatexCommand \ref{doppler} \end_inset ). The wavelength of the emitted radiation as observed in the laboratory frame is \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \lambda_{radiation}=\frac{\lambda_{w}}{2\gamma_{z}^{2}}\end{equation} \end_inset \layout Standard \paragraph_spacing double Here \begin_inset Formula $\gamma_{z}$ \end_inset is longitudinal kinetic energy of the electron where we have considered units of \begin_inset Formula $mc^{2}=1$ \end_inset . The total kinetic energy of the electron \begin_inset Formula $\gamma$ \end_inset can be expressed in terms of the longitudinal kinetic energy \begin_inset Formula $\gamma_{z}$ \end_inset as \begin_inset LatexCommand \cite{key-44} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \gamma^{2}=\gamma_{z}^{2}\left(1+\frac{K_{w}^{2}}{2}\right)\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $K_{w}=eB_{o}\lambda_{w}/(2\pi mc)$ \end_inset is the wiggler parameter. The wavelength of emitted radiation can thus be written as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \lambda_{radiation}=\lambda_{o}=\frac{\lambda_{w}}{2\gamma^{2}}\left(1+K_{w}^{2}\right)\end{equation} \end_inset \layout Standard \paragraph_spacing double This equation is known as the free electron laser \emph on resonance \emph default equation. This radiation is incoherent as the electrons are distributed randomly in their arrival time and their radiation adds incoherently. This incoherent radiation is termed as spontaneous emission. The spontaneous radiation provides the seed field for the generation of stimulated radiation. The spontaneous radiation is the synchrotron radiation produced by these accelerating electrons in the wiggler. The electrons radiate for a finite time, since the length of the wiggler is finite. This leads to the spectrum developing a Fourier transformed line width of \begin_inset Formula $\frac{1}{N_{w}}$ \end_inset where \begin_inset Formula $N_{w}$ \end_inset is the number of wiggler periods. \layout Standard \paragraph_spacing double It is worth noting that the electrons radiate even at higher harmonics in the wiggler. This implies that an electron beam in a wiggler maybe considered analogous to a conventional laser's non-linear medium. This property may be potentially used for producing shorter wavelength lasers by coherent harmonic generation. The intensity of radiation is confined to a small cone of \begin_inset Formula $\gamma\theta\simeq1,$ \end_inset as expected from synchrotron radiation. In a FEL the allowed angle is of the order of mrad, implying \begin_inset Formula $\theta\approx1/\gamma$ \end_inset , hence only the radiation along z axis is important. \layout Subsection \paragraph_spacing double Stimulated Emission \begin_inset LatexCommand \label{sub:Ch7_Stimulated-emission} \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel18_bunching.eps clip rotateOrigin center \end_inset \layout Caption Schematic of bunching in an FEL \begin_inset LatexCommand \label{cap:Schematic-of-bunching} \end_inset \layout Standard Electrons are shown in solid dots and the direction of the force is shown by arrows. As the sign of the ponderomotive force changes,the electrons get bunched at locations marked by b and anti-bunched at locations marked by a. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The electron beam traveling down the wiggler gets bunched as a result of the interaction with the co-propagating optical wave with frequency \begin_inset Formula $\omega_{r}$ \end_inset arising from the spontaneous \begin_inset Foot collapsed false \layout Standard In certain free electron lasers this radiation is provided by another seed laser \end_inset emission. This bunching may be explained in terms of the longitudinal Lorentz force that arises due to the coupling between transverse velocity component \begin_inset Formula $v_{x}$ \end_inset of the electron and the oscillating magnetic field \begin_inset Formula $\vec{B_{r}}$ \end_inset of the radiation field. This force is called ponderomotive force. The magnetic field of the linearised radiation field may be written as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \vec{B}_{r}=B_{ro}\cos(k_{r}z-\omega_{r}t+\phi_{r}t)\hat{y}\end{equation} \end_inset The electron velocity has components in the x and z directions \begin_inset LatexCommand \cite{key-47} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \vec{\beta}=\left[(-K_{w}/\gamma)\sin(k_{w}z),0,\beta_{z}\right]\end{equation} \end_inset The magnitude of the ponderomotive force \begin_inset Formula $F_{z}=ev_{x}B_{r}$ \end_inset is given by \begin_inset Formula \begin{equation} F_{z}\propto\left[-\sin(k_{r}z+k_{w}z-\omega_{r}t+\phi_{r})+\sin(k_{r}z-k_{w}z-\omega_{r}t+\phi_{r})\right]\end{equation} \end_inset The term \begin_inset Formula $(k_{r}z+k_{w}z-\omega_{r}t)$ \end_inset is called the ponderomotive phase, \begin_inset Formula $\Psi.$ \end_inset It can be shown that near the resonance condition, the z component of the electron velocity is \begin_inset Formula $\frac{\omega}{k_{r}+k_{w}}$ \end_inset . This makes the first term slowly varying and the second term rapidly oscillatin g between \begin_inset Formula $\pm1$ \end_inset along the length of the wiggler. Thus the contribution from the second term averages to zero and we are left with \begin_inset Formula \begin{equation} F_{z}\propto\sin(k_{r}z+k_{w}z-\omega_{r}t+\phi_{r})\end{equation} \end_inset Each electron, depending on its position, sees a different ponderomotive phase \begin_inset Formula $\Psi$ \end_inset and thus experiences a different magnitude and direction of ponderomotive force. This leads to bunching as shown in figure \begin_inset LatexCommand \ref{cap:Schematic-of-bunching} \end_inset \begin_inset LatexCommand \cite{key-49} \end_inset . The distance between successive bunch lengths is \begin_inset Formula $\frac{2\pi}{(k+k_{u})}\sim\lambda_{L}$ \end_inset . All the bunched electrons radiate with the same phase and successive bunches radiate with a phase difference of \begin_inset Formula $2\pi$ \end_inset . Hence this radiation develops coherence and is termed as stimulated emission. \layout Standard \paragraph_spacing double The FEL mechanism may be summarised in three steps: \layout Enumerate the electrons oscillate inside a wiggler and emit spontaneous radiation. \layout Enumerate this radiation acts back on the undulating electrons and bunches them. \layout Enumerate the bunched electrons emit coherent stimulated radiation and amplify the co-propagating electromagnetic wave. \layout Section \paragraph_spacing double Optical Klystron \begin_inset LatexCommand \label{Ch7_OK} \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel16_OpticalKlystronSchematic.eps clip rotateOrigin center \end_inset \layout Caption Schematic layout of optical klystrons in FELs. \begin_inset LatexCommand \label{cap:Schematic-OK} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double Beams of relativistic electrons used for FEL operations can be produced by a variety of accelerators. The most popular means are RF accelerators, storage rings and linear accelerato rs using photo-injector technology. We discuss storage ring FELs as this thesis reports on simulations and experiments conducted at the 1.2 GeV Duke storage ring based free electron laser. \layout Standard \paragraph_spacing double All operational storage ring free electron lasers are based on a scheme of optical klystron (OK) as suggested by Vinokurov and Skrinsky \begin_inset LatexCommand \citet{key-26} \end_inset . An optical klystron is a modification of the conventional free electron laser and is designed to improve the FEL gain \begin_inset Foot collapsed false \layout Standard Gain is defined as \begin_inset Formula $G_{in}=\frac{(P_{out}-P_{in})}{P_{in}}$ \end_inset where \begin_inset Formula $P_{in}$ \end_inset is the input power and \begin_inset Formula $P_{out}$ \end_inset is the output power. \end_inset . An optical klystron consists of two wigglers separated by a bunching section (also called dispersive section) consisting of one period of deflecting magnets which provides for optimal bunching conditions for the e-beam. A schematic sketch of an optical klystron is shown in figure \begin_inset LatexCommand \ref{cap:Schematic-OK} \end_inset . In an optical klystron the function of modulation and of radiation are separated. The optical wave creates energy modulation of the electron beam in the first undulator at the optical spatial period of \begin_inset Formula $\lambda_{r}$ \end_inset of the field. In the dispersive section this energy modulation is transformed into density modulation. This can be understood as illustrated by figure \begin_inset LatexCommand \ref{FEL_EbeamPhase} \end_inset . This figure shows Poincare plots in the \begin_inset Formula $(\varphi,\epsilon/E_{o})$ \end_inset space. The electrons enter the first wiggler with some initial spread in energy and random phases as seen in \begin_inset LatexCommand \ref{FEL_EbeamPhase} \end_inset (a). In transit through the first wiggler, electrons lose or gain energy depending on the phase between electron motion and the optical field. This leads to an energy modulation, as depicted in figure \begin_inset LatexCommand \ref{FEL_EbeamPhase} \end_inset (b). As the electrons pass through the buncher, the more energetic electrons catch up with the less energetic electrons in front, thus converting the energy modulation into a density modulations seen in figure \begin_inset LatexCommand \ref{FEL_EbeamPhase} \end_inset (c). This modulated beam interacts with the optical wave in the second wiggler, and if the phase delay between the two wigglers is chosen correctly, most electrons entering the second wiggler will lose energy and transfer it to the optical field, thus amplifying the optical beam as in figure \begin_inset LatexCommand \ref{FEL_EbeamPhase} \end_inset (d). As figure \begin_inset LatexCommand \ref{FEL_EbeamPhase} \end_inset (e) shows, after one revolution, the bunching on the scale of optical wavelengt h is washed out due to very large spread in the pass around time of the individual electrons.The washing out of the correlation causes growth of the energy spread in electron beam, which saturates FEL gain and average FEL power in storage rings. \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel17_phaseSpace.eps display none rotateOrigin center \end_inset \layout Caption Evolution of electron in the phase space of \begin_inset Formula $\varphi$ \end_inset and \begin_inset Formula $\frac{\epsilon}{E}$ \end_inset \begin_inset LatexCommand \label{FEL_EbeamPhase} \end_inset \layout Standard (a)Shows the initial distribution of electron beam. (b) Shows the energy modulated electron beam in the first wiggler. (c) Shows the density modulated distribution of the e-beam in the buncher. (d) Shows the distribution of the e-beam in the second wiggler, where the coherent radiation develops. (e) Shows the e-beam after one complete revolution where the bunching is washed out. All the phase space diagrams are plotted in \begin_inset Formula $(\varphi,\frac{\epsilon}{E})$ \end_inset space, where \begin_inset Formula $\epsilon=E-E_{o}$ \end_inset is the energy deviation of the electron and \begin_inset Formula $\varphi$ \end_inset is the phase of electrons. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double The power spectrum of the spontaneous emission from an electron travelling in the wiggler can be calculated from the Lienard -Wiechart potentials. In the far field, the power spectrum in the frequency range \begin_inset Formula $d\omega$ \end_inset and the solid angle \begin_inset Formula $d\Omega$ \end_inset about the direction of observation \begin_inset Formula $\hat{n}$ \end_inset is \begin_inset LatexCommand \cite{key-37} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \frac{d^{2}I}{d\omega d\Omega}=\frac{e^{2}\omega^{2}}{4\pi^{2}c}\left|\int dt\hat{n}\times(\hat{n}\times\vec{\beta(t)})exp\left[i\omega\left(t-\frac{\hat{n}\centerdot\vec{r(t)}}{c}\right)\right]\right|^{2}\end{equation} \end_inset \layout Standard \paragraph_spacing double In an optical klystron, the wave packet of spontaneous radiation from the second wiggler is delayed by time duration corresponding to the total slippage \begin_inset Formula $(\bigtriangleup s)$ \end_inset of electrons. Slippage is defined as the path difference along the axis of propagation of the e-beam (z axis) between the optical pulses travelling at speed of light and the electron travelling at comparable but less than speed of light. Neglecting the radiation from the buncher, which is much weaker and broader than the radiation from the wigglers, the power spectrum of the OK spontaneous radiation may be written as \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \left(\frac{d^{2}I}{d\omega d\Omega}\right)_{OK}=2.\left(\frac{d^{2}I}{d\omega d\Omega}\right)_{N_{w}}\left(1+\cos\left(k\bigtriangleup s\right)\right)\end{equation} \end_inset \layout Standard \paragraph_spacing double where \layout Standard \paragraph_spacing double \begin_inset Formula \begin{eqnarray} \left(\frac{d^{2}I}{d\omega d\Omega}\right)_{N_{w}} & = & \frac{e^{2}\omega^{2}}{4\pi^{2}c^{3}}\left(\frac{K_{w}\left[J_{o}(\zeta)-J_{1}(\zeta)\right]L_{w}}{2\gamma}\right)^{2}\left(\frac{\sin\left(\delta KL_{w}/2\right)}{\left(\delta KL_{w}/2\right)}\right)^{2}\\ \delta K & = & \left(\frac{k}{2\gamma^{2}}\left(1+\frac{K_{w}^{2}}{2}\right)-k_{w}\right)\\ \zeta & = & \frac{1}{2}\frac{K_{w}^{2}/2}{\left(1+\mathbf{\mathnormal{K_{w}^{2}}/\mathnormal{2}}\right)}\end{eqnarray} \end_inset Here \begin_inset Formula $\bigtriangleup s$ \end_inset is the slippage between the centers of the two wigglers, \begin_inset Formula $N_{d}$ \end_inset is dimensionless parameter characterising the buncher, \begin_inset Formula $k_{w}$ \end_inset is the wave vector of the wiggler and \begin_inset Formula $k=\frac{\omega}{c}$ \end_inset . The subscript \begin_inset Formula $N_{w}$ \end_inset indicates spectrum from one wiggler with \begin_inset Formula $N_{w}$ \end_inset periods and the subscript OK denotes the spectrum from an optical klystron with two wigglers and a buncher. \layout Section \paragraph_spacing double Duke Storage Ring Free Electron laser \begin_inset LatexCommand \label{sec:Ch7Duke-Storage-Ring} \end_inset \layout Standard \paragraph_spacing double The Duke storage ring is designed to drive ultraviolet and soft X-ray free electron lasers. One of the two 34 m straight section is dedicated to FEL installation. The other straight section is used for installation of RF cavity and diagnostic systems. The Duke storage ring employs a linear accelerator (linac) as its injector and a thermionic RF gun as its electron source. The electron pulse is generated by a photo-gun. The present linac delivers up-to 270 MeV electron beams to the storage ring. The injected beam can be stored in the ring and accelerated to a desirable energy of operation up to \begin_inset Formula $1.2$ \end_inset GeV. A schematic sketch of the storage ring layout is given in figure \begin_inset LatexCommand \ref{OK4-FEL} \end_inset with a list of design parameters in table \begin_inset LatexCommand \ref{cap:tableDukeFel} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel2_StorageRing.eps display none clip \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \layout Caption Schematic layout of the OK-4/Duke SRFEL \begin_inset LatexCommand \label{OK4-FEL} \end_inset \end_inset \layout Standard \begin_inset Float table wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard \series bold Storage Ring \end_inset \begin_inset Text \layout Standard \series bold Design Parameters \end_inset \begin_inset Text \layout Standard Operating Energy [GeV] \end_inset \begin_inset Text \layout Standard 0.25-1 \end_inset \begin_inset Text \layout Standard Injection Energy [GeV] \end_inset \begin_inset Text \layout Standard 0.27 \end_inset \begin_inset Text \layout Standard Ring Circumference [m] \end_inset \begin_inset Text \layout Standard 107.46 \end_inset \begin_inset Text \layout Standard Arc Section [m] \end_inset \begin_inset Text \layout Standard 19.52 \end_inset \begin_inset Text \layout Standard Straight Section [m] \end_inset \begin_inset Text \layout Standard 34.21 \end_inset \begin_inset Text \layout Standard Revolution Frequency \end_inset \begin_inset Text \layout Standard 2.78 \end_inset \begin_inset Text \layout Standard RF Frequency \end_inset \begin_inset Text \layout Standard 178.5 \end_inset \begin_inset Text \layout Standard Number of Dipoles \end_inset \begin_inset Text \layout Standard 40 \end_inset \begin_inset Text \layout Standard Number of Quadrapoles \end_inset \begin_inset Text \layout Standard 64 \end_inset \begin_inset Text \layout Standard \series bold Electron Beam \end_inset \begin_inset Text \layout Standard \series bold Design Parameters \end_inset \begin_inset Text \layout Standard Beam Current [A] \end_inset \begin_inset Text \layout Standard 0.1 \end_inset \begin_inset Text \layout Standard Peak Current [A] \end_inset \begin_inset Text \layout Standard 30-50 \end_inset \begin_inset Text \layout Standard Horizontal Emittance [m-mrad] \end_inset \begin_inset Text \layout Standard 18 \begin_inset Formula $\times10^{-9}$ \end_inset \end_inset \begin_inset Text \layout Standard Vertical Emittance [m-mrad] \end_inset \begin_inset Text \layout Standard 1 \begin_inset Formula $\times10^{-9}$ \end_inset \end_inset \begin_inset Text \layout Standard Bunch Length [ps] \end_inset \begin_inset Text \layout Standard 33 \end_inset \end_inset \layout Caption Design parameters of Duke storage ring FEL. \begin_inset LatexCommand \label{cap:tableDukeFel} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Subsection \paragraph_spacing double OK-4/Duke SRFEL \layout Standard \paragraph_spacing double The OK-4 magnetic system is comprised of two 3.3 meter long electromagnetic wigglers and a buncher located between them. As mentioned earlier the use of a buncher distinguishes an optical klystron from a conventional free electron laser and is used to optimise gain by optimizing longitudinal dispersion. The main parameters of the OK-4 at Duke are listed in table \begin_inset LatexCommand \ref{OK4Parameters} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float table wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard \series bold Wiggler \end_inset \begin_inset Text \layout Standard \end_inset \begin_inset Text \layout Standard \series bold Buncher \end_inset \begin_inset Text \layout Standard \end_inset \begin_inset Text \layout Standard Length [m] \end_inset \begin_inset Text \layout Standard 3.3 \end_inset \begin_inset Text \layout Standard Length [m] \end_inset \begin_inset Text \layout Standard 0.34 \end_inset \begin_inset Text \layout Standard Peak Magnetic Field [KGs] \end_inset \begin_inset Text \layout Standard 0-5.8 \end_inset \begin_inset Text \layout Standard Magnetic Field [KGs] \end_inset \begin_inset Text \layout Standard 0-12 \end_inset \begin_inset Text \layout Standard Period [m] \end_inset \begin_inset Text \layout Standard 0.10 \end_inset \begin_inset Text \layout Standard \end_inset \begin_inset Text \layout Standard \end_inset \begin_inset Text \layout Standard Wiggler parameter \begin_inset Formula $K_{w}$ \end_inset \end_inset \begin_inset Text \layout Standard 0-5.42 \end_inset \begin_inset Text \layout Standard \end_inset \begin_inset Text \layout Standard \end_inset \end_inset \layout Caption Main parameters of Duke OK-4 magnetic system \begin_inset LatexCommand \label{OK4Parameters} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section \paragraph_spacing double Giant Pulses at OK4/Duke SR FEL \begin_inset LatexCommand \label{sec:Ch7_Giant-Pulses-at} \end_inset \layout Standard \paragraph_spacing double For high power applications the average storage ring FEL power can be redistribu ted into a series of giant pulses. A giant pulse is a very high peak power macro pulse, comprising of many micro-pulses, within a smooth envelope. Typically a macro-pulse at Duke SR FEL comprises of 100 micro pulses separated by e-beam revolution time of 350 ns. The various techniques to obtain giant pulses are : \layout Enumerate conventional Q modulation technique \begin_inset LatexCommand \cite{key-30} \end_inset \layout Enumerate the gain modulation technique \begin_inset LatexCommand \cite{key-31} \end_inset \layout Enumerate RF frequency modulation called the Q switching technique \begin_inset LatexCommand \cite{key-32} \end_inset \layout Standard \paragraph_spacing double At Duke OK-4/SR FEL the gain modulation technique is used \begin_inset LatexCommand \cite{key-33} \end_inset . In this technique the lasing is stopped by moving the e beam orbit away from the axis of the optical cavity typically for \begin_inset Formula $X_{o}=2.5$ \end_inset \begin_inset Formula $mm$ \end_inset in the horizontal direction. This distortion of the orbit is shown in figure \begin_inset LatexCommand \ref{cap:Ch7_OrbitDistortion} \end_inset . Returning the orbit to the axis, adiabatically, generates a giant pulse. This technique of moving and returning the e-beam orbit onto the optical axis is implemented by using a fast steering magnet, called the gain modulator, driven by a critically damped circuit with a time constant \begin_inset Formula $\tau=8.8$ \end_inset \begin_inset Formula $\mu s$ \end_inset . This time is much shorter than the time duration of the giant pulse which is typically 50 \begin_inset Formula $\mu s$ \end_inset to 100 \begin_inset Formula $\mu s$ \end_inset as can be seen in figure \begin_inset LatexCommand \ref{GiantPulseOrbit} \end_inset . \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel3_OrbitDistortion.eps clip \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \layout Caption The orbit distortion in the electron beam created by the gain modulator. \begin_inset LatexCommand \label{cap:Ch7_OrbitDistortion} \end_inset \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel4_GiantPulseTime.eps display none clip \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \layout Caption Orbit transition of the electron beam during giant pulse generation \begin_inset LatexCommand \label{GiantPulseOrbit} \end_inset \end_inset \layout Standard \paragraph_spacing double In the giant pulse mode of operation of the OK-4/Duke SR FEL the gain modulator periodically moves the closed electron beam orbit in the FEL region away from the axis of the optical cavity. This stops lasing and allows the electron beam energy spread to be reduced to its original value. A pulse generator, defines the sequence of the giant pulses with desirable repetition rate \begin_inset Formula $f_{rep}$ \end_inset . Each pulse from the generator initiates adiabatic transition of fresh electrons beam onto optical axis and a giant FEL pulse is generated. The temporal profile of such a pulse from experimental data at OK-4/Duke SRFEL is shown in Figure \begin_inset LatexCommand \ref{GianPulseTemporal} \end_inset . \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel6_GiantPulseTemporalProfile.eps display none clip \end_inset \layout Caption Temporal profile of giant pulses created by gain modulation at OK-4/Duke SRFEL. \begin_inset LatexCommand \label{GianPulseTemporal} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset A summary of the parameters obtained in the giant pulse mode is given in table \begin_inset LatexCommand \ref{cap:Ch7_GiantPulseModeTable} \end_inset . \layout Standard \paragraph_spacing double \begin_inset Float table wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard E-beam energy (in GeV) \end_inset \begin_inset Text \layout Standard 0.25-0.8 \end_inset \begin_inset Text \layout Standard Wavelength (fundamental in nm) \end_inset \begin_inset Text \layout Standard 194-730 \end_inset \begin_inset Text \layout Standard Giant pulse rep rate (in Hz) \end_inset \begin_inset Text \layout Standard 1-60 \end_inset \begin_inset Text \layout Standard Macro-pulse energy (in mJ) \end_inset \begin_inset Text \layout Standard 0.05-3 \end_inset \begin_inset Text \layout Standard Macro-pulse duration (FWHM in \begin_inset Formula $\mu s$ \end_inset ) \end_inset \begin_inset Text \layout Standard 40-200 \end_inset \begin_inset Text \layout Standard Peak Intra-cavity power (in GW) \end_inset \begin_inset Text \layout Standard 0.1-1.2 \end_inset \begin_inset Text \layout Standard Line Width (FWHM \begin_inset Formula $\frac{\delta\lambda}{\lambda}$ \end_inset ) \end_inset \begin_inset Text \layout Standard \begin_inset Formula $(1-2)\times10^{-4}$ \end_inset \end_inset \begin_inset Text \layout Standard Micro-pulse duration ( \begin_inset Formula $\sigma\tau$ \end_inset in ps) \end_inset \begin_inset Text \layout Standard 15-25 \end_inset \begin_inset Text \layout Standard Micro-pulse separation (in ns) \end_inset \begin_inset Text \layout Standard 358.4-5.60 \end_inset \begin_inset Text \layout Standard Spatial distribution \end_inset \begin_inset Text \layout Standard Primarily \begin_inset Formula $TEM_{oo}$ \end_inset \end_inset \end_inset \layout Caption OK-4/Duke SRFEL in giant pulse mode \begin_inset LatexCommand \label{cap:Ch7_GiantPulseModeTable} \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double One of the most important achievements of the OK-4/Duke SR FEL gain modulator is the generation of intra-cavity peak power in the GW level in deep UV part of electromagnetic spectrum. This level of power is at the level of super-pulse \begin_inset LatexCommand \cite{key-34} \end_inset and is sufficient for strong bunching at harmonics of the fundamental wavelengt h \begin_inset LatexCommand \cite{key-35} \end_inset . \layout Subsection \paragraph_spacing double Simulation of Giant Pulses at Duke OK4/SR-FEL \begin_inset LatexCommand \label{sub:Ch7_Simulation-of-Giant} \end_inset \layout Standard \paragraph_spacing double A particle tracking code which includes all phenomenon influencing operation of short-wavelength storage ring FELs (SRFEL) was developed by V.N.Litvinenko et al \begin_inset LatexCommand \cite{key-39} \end_inset . The model used in the code simulates an optical klystron with total length 2L comprised of two planar wigglers,each having \begin_inset Formula $N_{w}$ \end_inset periods of \begin_inset Formula $\lambda_{w}$ \end_inset wavelength and a buncher occupying a length from \begin_inset Formula $-z_{i}$ \end_inset to \begin_inset Formula $z_{i}$ \end_inset . This set of high efficiency algorithms used in the code simulates real time self consistent SRFEL operation by tracking up to 1,000,000 macro-particle s for tens of thousands turns. \layout Subsection \paragraph_spacing double Concept of the Algorithm \begin_inset LatexCommand \label{sub:Ch7_Algo} \end_inset \layout Standard \paragraph_spacing double \series bold \emph on \noun on Macro-particles \layout Standard \paragraph_spacing double A macro-particle is described by its energy \begin_inset Formula $\epsilon=E-E_{o}\setminus E_{o}$ \end_inset and longitudinal position \begin_inset Formula $s=ct_{o}$ \end_inset where \begin_inset Formula $t_{o}$ \end_inset is the electron arrival time at \begin_inset Formula $z=0$ \end_inset . It represents a cluster of electrons localized in \begin_inset Formula $(\epsilon,s)$ \end_inset phase space and having standard distribution in the transverse phase space. The radiation is such a macro-particle can be found analytically \begin_inset LatexCommand \cite{key-36} \end_inset . This makes it possible to simulate the SRFEL operation. The maximum number of macro-particles in the code is set at 1,000,000. \layout Standard \paragraph_spacing double \series bold Optical Wave-packet \layout Standard \paragraph_spacing double The FEL optical field \begin_inset Formula $\vec{E}$ \end_inset is represented by a \begin_inset Formula $TEM_{oo}$ \end_inset mode with a slowly varying complex amplitude \begin_inset Formula $A_{o}(s)$ \end_inset . \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} E_{o}(t,\vec{r})=Re\left\{ \hat{e_{x}}A_{o}(s)\frac{\beta_{o}}{\beta-iz}exp\left[iks-\frac{k}{2}\frac{x^{2}+y^{2}}{\beta_{o}-iz}\right]\right\} \end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $s=ct-z$ \end_inset and c is the speed of light. The Rayleigh length is \begin_inset Formula $\beta_{o}$ \end_inset with waist at azimuth \begin_inset Formula $z=0$ \end_inset , x and y are transverse coordinates and \begin_inset Formula $k=2\pi/\lambda$ \end_inset is the optical wave vector. An optical wave packet is represented by an array of complex wave amplitudes \begin_inset Formula $A[i]$ \end_inset . \layout Standard \paragraph_spacing double \series bold Slippage \layout Standard \paragraph_spacing double The induced radiation (both real and imaginary part) is shifted with respect to the optical wave-packet. This shift depends on the electron energy and is essential for the optical pulse evolution as it influences the super-mode structure. The slippage is modelled by taking the average as well as the first moment of the real and imaginary part of the gain. \layout Subsection \paragraph_spacing double Simulations Code \begin_inset LatexCommand \label{sub:Ch7Simulations-Code} \end_inset \layout Standard \paragraph_spacing double The program comprises of four main parts \layout Standard \paragraph_spacing double \series bold \emph on FelCoefficients- \series default \emph default Reads tabulated integrals and calculates arrays of real and imaginary values of macro-particle gain and moments. \layout Standard \paragraph_spacing double \series bold \emph on RingPass- \series default \emph default It has the algorithm for round trip pass of the macro-particles \series bold \emph on \series default \emph default around the storage ring \series bold \emph on . \layout Standard \paragraph_spacing double \series bold \emph on FELInteraction- \series default \emph default It is the main part of the code. It finds the two wave-packet bins the particle is interacting with and calculates the optical field using the small signal approximation. It also calculates the energy bin corresponding to the instantaneous macro-part icle energy and adds the contribution of the spontaneous radiation from the correct energy bin into the wave-packet bins. This contribution is proportional to the overlap as shown in figure \begin_inset LatexCommand \ref{Overlap} \end_inset . The induced radiation is calculated using complex gain \begin_inset Formula $g_{av}$ \end_inset and its first moment \begin_inset Formula $g_{moment}$ \end_inset and the optical wave amplitude \begin_inset Formula $A_{o}$ \end_inset : \layout Standard \paragraph_spacing double \begin_inset Formula \begin{eqnarray} A_{o} & = & A_{n}[i]\left(1-X\right)+A_{n}[i+1]X\\ A_{n+1}[i] & = & A_{n}[i]+A_{o}(1-X)\left\{ g_{av}\left[e_{bin}\right]-g_{moment}\left[e_{bin}\right]X\right\} \\ A_{n+1}[i+1] & = & A_{n}[i+1]+A_{o}X\left\{ g_{av}\left[e_{bin}\right]-g_{moment}\left[e_{bin}\right]\left(1-X\right)\right\} \end{eqnarray} \end_inset \layout Standard \paragraph_spacing double where X is the overlap of macro-particle radiation and wave packet bins,i is the label running over the wave-packet bins and n is the label running over iterations in the code. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel7_Overlap.eps \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \layout Caption Overlap of optical wave and the electron bin. \begin_inset LatexCommand \label{Overlap} \end_inset \end_inset \layout Standard \paragraph_spacing double \series bold \emph on CavityPass \series default \emph default -Implements the exact transformation described by: \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} A_{n+1}(s)=\sqrt{R}A_{n}(s-d)\end{equation} \end_inset \layout Standard \paragraph_spacing double where R is the round trip reflectivity and d is the detuning of the optical cavity. It accumulates the total value of the detuning from pass to pass and ensures that the wave-packet moves only an integral part of the accumulated detuning. The non integral part is sent to the subroutine \emph on FELInteraction \emph default , where it is accounted for in the process of interaction. \layout Standard \paragraph_spacing double Simulations of giant pulses by this self consistent storage ring FEL code has been used to project and later confirm the performance of the OK-4/Duke SRFEL. The peak power as simulated by the code as a function of the turn number if shown in figure \begin_inset LatexCommand \ref{cap:Ch7_PeakPower} \end_inset . \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel9_GiantpulsePower.eps clip \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \layout Caption Peak power in giant pulses. \begin_inset LatexCommand \label{cap:Ch7_PeakPower} \end_inset \layout Standard On the y axis is plotted on a log scale the increase in peak power(red) and energy(blue) in the optical pulse. \end_inset \layout Standard \paragraph_spacing double A significant upshot of these simulations was the discovery of the super-pulse regime. These simulations showed that while the energy in the micro-pulses corresponds to earlier theoretical predictions, the peak power turns out to be 10 times higher than expected. This useful discrepancy is caused by super pulse phenomenon and makes coherent harmonic generation possible in storage rings without using external drive lasers (See \begin_inset LatexCommand \ref{sec:Ch7_Harmonics} \end_inset ) . \layout Subsection \paragraph_spacing double Electron-Beam Evolution \begin_inset LatexCommand \label{sub:Ch7_Electron-beam-evolution} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel8_ebeamEvolution.eps display none clip \end_inset \layout Caption Evolution of the e-beam simulated by self consistent three dimensional SRFEL code \begin_inset LatexCommand \label{FELHistogram} \end_inset \layout Standard A histogram of number of particles vs position is plotted for various turn numbers. This captures the evolution of the electron beam. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double We use the code further to study the evolution of the electron beam in the giant pulse mode. To follow the evolution of the e-beam the macro-particles are tagged. As each macro-particles is defined by \begin_inset Formula $\bigtriangleup E=(E-E_{o})/E_{o}$ \end_inset and an \begin_inset Formula $s=ct-z$ \end_inset , we tag the \begin_inset Formula $\bigtriangleup E(m)$ \end_inset and \begin_inset Formula $s(m)$ \end_inset where m runs from 1 to a maximum of 1,000,000. At the end of each revolution or any arbitrary number of revolutions, the values of \begin_inset Formula $\bigtriangleup E$ \end_inset and \begin_inset Formula $s$ \end_inset are noted for each m. A histogram, of the number of macro-particles vs the longitudinal distance is sufficient to depict the evolution of the e-beam. Figure \begin_inset LatexCommand \ref{FELHistogram} \end_inset shows the evolution of the electron beam over various revolution of the storage ring. Such evolution of the electron beam were studied by simulations and compared with experimental results as restored by phase space tomography techniques \begin_inset LatexCommand \cite{key-54} \end_inset . Such a comparison is shown in Figure \begin_inset LatexCommand \ref{CompareKevin} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel10_comparison.eps clip \end_inset \layout Caption Comparison of simulation vs.reconstructed experimental data. \begin_inset LatexCommand \label{CompareKevin} \end_inset \layout Standard This figure shows the evolution of the electron beam in the \begin_inset Formula $(\delta,\zeta)$ \end_inset phase space, where \begin_inset Formula $\delta=\epsilon/\sigma_{\epsilon}$ \end_inset is the energy deviation normalised by initial rms value of the energy deviation and \begin_inset Formula $\zeta=z/\sigma_{l}$ \end_inset is the z normalised by the optical bunch length. (a) is obtained by the simulation code and (b) is obtained by reconstruction of streak camera images with phase space tomography techniques. Both (a) and (b) are comparisons of the e-beam phase space at 255th revolution in the storage ring. (c) shows a comparison of the two dimensional profile as obtained by simulation s and the reconstruction method. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section \paragraph_spacing double Harmonic Generation at OK-4/Duke SR FEL \begin_inset LatexCommand \label{sec:Ch7_Harmonics} \end_inset \layout Subsection \paragraph_spacing double Theory of Harmonic Generation at SR FELS \layout Standard \paragraph_spacing double It has been discussed in the operation of the optical klystron (Refer \begin_inset LatexCommand \ref{Ch7_OK} \end_inset ) that the interaction of the electron with the optical filed in the first wiggler produces energy modulation. The buncher transforms this energy modulation into density modulation with a period equal to the optical wavelength \begin_inset Formula $\lambda_{o}$ \end_inset . The density modulation has all higher harmonics of the fundamental wavelength present in it. The electron beam radiates coherently into second wiggler. The coherent radiation from the second wiggler can be connected to the spontaneous radiation by the following relation \begin_inset LatexCommand \cite{key-50} \end_inset \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} P_{N,coherent}=\tilde{n_{e}}P_{N,spontaneous}B_{N}^{2}ff_{N}\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $\tilde{n_{e}}$ \end_inset is the effective number of electrons participating in the process of coherent radiation, bunching efficiency \begin_inset Formula $B_{N}$ \end_inset is bunching efficiency and \begin_inset Formula $ff_{N}$ \end_inset is a form factor which includes optical cavity geometry, e-beam emittance and energy spread. The bunching efficiency is defined as \begin_inset Formula \begin{equation} B_{N}\equiv\left|\frac{\rho_{N}}{\rho_{o}}\right|\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $\rho_{N}$ \end_inset is the harmonic content of the current density at the exit of the buncher and is given by \layout Standard \paragraph_spacing double \begin_inset Formula \begin{equation} \rho_{n}=\int_{0}^{2\pi}\rho(z)\exp(-in\phi_{b})d\phi_{b}=\int_{-\infty}^{\infty}d\epsilon\int_{0}^{2\pi}d\phi_{o}f(\epsilon_{o})\exp(\phi_{b})\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $\phi_{b}$ \end_inset is the expression for phase of an electron of initial energy E when it reaches the end of the buncher and is given by \begin_inset Formula \begin{equation} \phi_{b}=\phi_{o}+4\pi N_{d}(\epsilon_{o}+\bigtriangleup\epsilon_{mod}\cos\phi_{o})-2\pi N_{d}\end{equation} \end_inset \layout Standard where \begin_inset Formula $\bigtriangleup\epsilon_{mod}$ \end_inset is the energy modulation \begin_inset LatexCommand \cite{key-41} \end_inset given by \begin_inset Formula \begin{equation} \bigtriangleup\epsilon_{mod}=\frac{N_{w}\lambda_{w}a_{w}}{\gamma}\left|e\vec{E_{o}}\right|jj\end{equation} \end_inset \layout Standard In formula 6.25 \begin_inset Formula $jj=(J_{o}(\zeta)-J_{1}(\zeta))/\sqrt{2}$ \end_inset , \begin_inset Formula $\zeta=a_{w}^{2}/1+a_{w}^{2}$ \end_inset and \begin_inset Formula $a_{w}=K_{w}/\sqrt{2}$ \end_inset for a planar wiggler. \layout Standard \paragraph_spacing double Assuming a Gaussian distribution of the electrons, \begin_inset Formula $f(\epsilon_{o})$ \end_inset can be substituted as \begin_inset Formula $f(\epsilon_{o})=\frac{\rho_{o}}{\sqrt{2\pi}\sigma_{\epsilon}}\exp(-\frac{\epsilon_{o}}{2\sigma_{\epsilon}^{2}})$ \end_inset , t he integral then becomes \begin_inset Formula \begin{equation} \rho_{n}=\rho_{o}J_{N}(nX)\exp(\frac{(-4\pi N_{d}\sigma_{\epsilon}n/E)^{2}}{2})\end{equation} \end_inset \layout Standard \paragraph_spacing double where \begin_inset Formula $X=4\pi N_{d}\bigtriangleup\epsilon_{mod}$ \end_inset and \begin_inset Formula $J_{N}$ \end_inset is the ordinary \begin_inset Formula $n^{th}$ \end_inset order Bessel function of the first kind. Thus the bunching efficiency term reduces to \begin_inset Formula \begin{equation} B_{N}=J_{N}(nX)\exp(\frac{(-4\pi N_{d}\sigma_{\epsilon}n)^{2}}{2})\end{equation} \end_inset \layout Standard \paragraph_spacing double Thus the power of the coherent radiation depends strongly on the energy spread \begin_inset LatexCommand \cite{key-41} \end_inset \begin_inset Formula \begin{equation} P_{N,coherent}\sim\exp(-4\pi N_{d}\sigma_{\epsilon}n)^{2}\end{equation} \end_inset \layout Standard \paragraph_spacing double Finite energy spread of the e-beam at \begin_inset Formula $\sim0.1\%$ \end_inset sets the limitation on the harmonic number at \begin_inset Formula $n\sim20$ \end_inset . \layout Standard \paragraph_spacing double The generation of harmonics is made possible at OK-4/Duke SRFEL by operating in the super pulse mode which has GW level intra-cavity peak power. The e-beam dynamics in this mode provides for refreshing part of the electron beam during the generation of the giant pulse \begin_inset LatexCommand \cite{key-52} \end_inset . This fresh electron beam has natural spread and leads to effective harmonic generation. \layout Subsection \paragraph_spacing double Harmonic Generation Experiments \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel11_harmonicSchematic.eps clip \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \layout Caption Experimental Schematic for the harmonic generation experiment at OK-4/Duke SRFEL. \begin_inset LatexCommand \label{ExpSetupHarmonic} \end_inset \end_inset \layout Standard \paragraph_spacing double A proof of principle experiment was performed at OK-4/Duke SR-FEL to test the feasibility of harmonic generation in storage ring FELs without using external drive lasers. The experiment was performed using visible mirrors, partially transparent in deep-UV part of the electromagnetic spectrum. In preparation of this experiment a UHV beam line equipped with focusing Ir-coated mirror as well as mirrors with 0.5 mm diameter hole in the center were manufactured, developed and tested. This was deemed necessary as there is no available transparent optics below \begin_inset Formula $115$ \end_inset \begin_inset Formula $nm$ \end_inset and wavelengths expected from harmonic generation were well below this mark. The holes in the mirror provided for in-vacuum out-coupling of the harmonics while introducing an additional \begin_inset Formula $\sim1$ \end_inset % losses in the cavity. \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel15_mode.eps clip \end_inset \layout Caption Lasing with a holed cavity mirror. \begin_inset LatexCommand \label{HoledMirros} \end_inset \layout Standard (a) shows the mirror with a 0.5 mm diameter hole in it. Transparent optics is not available below 115 \begin_inset Formula $nm$ \end_inset and the harmonics generated were in the range of 86-37 \begin_inset Formula $nm$ \end_inset , and a hole was made in the downstream mirror to out-couple the radiation generated by the harmonics of the fundamental lasing wavelength. (b) Shows the speckle pattern created on a phosphor screen at the fundamental lasing wavelength of 260 \begin_inset Formula $nm$ \end_inset . This pattern was generated due to the change in the geometry of the optical cavity by the hole. Analysis showed that most of the power continued to be in the \begin_inset Formula $TEM_{oo}$ \end_inset mode. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset Though the hole in the mirror changes the geometry of the cavity most of the power is still sustained in \begin_inset Formula $TEM_{oo}$ \end_inset mode. The diagnostics for this experiment also included a Nova VUV UHV monochromator \begin_inset Foot collapsed false \layout Standard Manufactured by McPherson Inc. \end_inset which can operate down to 30 nm. A schematic of the diagnostic set up is shown in figure \begin_inset LatexCommand \ref{ExpSetupHarmonic} \end_inset . \layout Subsubsection \paragraph_spacing double Results from Harmonic Generation Experiment \layout Standard \paragraph_spacing double The wavelength of the lasing was tunable from 240 nm to 700 nm and intense coherent harmonic pulses were generated using 460 MeV e-beam. Coherent harmonics were observed at the second (118-139 \begin_inset Formula $nm$ \end_inset ), third (79-97 \begin_inset Formula $nm$ \end_inset ), fourth (59-70 \begin_inset Formula $nm$ \end_inset )and fifth (49-55 \begin_inset Formula $nm$ \end_inset ) and seventh ( \begin_inset Formula $37$ \end_inset \begin_inset Formula $nm$ \end_inset ) shown in figures \begin_inset LatexCommand \ref{Harmonic3} \end_inset , \begin_inset LatexCommand \ref{Harmonic5and7} \end_inset respectively. As the OK-4/FEL is a system with high K planar wigglers, all odd harmonics are generated on axis and the even harmonic are generated off axis and have less intensity. Preliminary analysis of the third harmonic at 86 nm showed an increased peak brightness by \begin_inset Formula $\sim2.5\times10^{5}$ \end_inset and peak power \begin_inset Formula $\sim60KW$ \end_inset . Detailed analysis of the results will be published elsewhere \begin_inset LatexCommand \cite{key-53} \end_inset . \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel12_ThirdHarmonic.eps display none clip \end_inset \layout Caption Third harmonic at 86.1 \begin_inset Formula $nm$ \end_inset of the fundamental lasing wavelength at 268 \begin_inset Formula $nm.$ \end_inset \begin_inset LatexCommand \label{Harmonic3} \end_inset \layout Standard Intensity in arbitrary units vs time is plotted. Red curve shows power at the fundamental wavelength while the blue curve shows power in the third harmonic. \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset ERT status Open \layout Standard \backslash begin{center} \end_inset \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/fel14_Harmonic5And7.eps display none clip \end_inset \layout Caption Detection of \begin_inset Formula $5^{th}$ \end_inset and \begin_inset Formula $7^{th}$ \end_inset harmonic. \begin_inset LatexCommand \label{Harmonic5and7} \end_inset \layout Standard (a) shows the generation of the 5th harmonic of the fundamental wavelength of \begin_inset Formula $250$ \end_inset \begin_inset Formula $nm$ \end_inset . (b) shows the presence of the 7th harmonic of the fundamental wavelength of 250 \begin_inset Formula $nm$ \end_inset . \layout Standard \begin_inset ERT status Open \layout Standard \backslash end{center} \end_inset \end_inset \layout Section Summary \begin_inset LatexCommand \label{sec:Ch7_Summary} \end_inset \layout Standard \paragraph_spacing double In this chapter we discuss photon beams produced by free electron lasers. The first couple of sections of the chapter provide a review of the principle of free electron lasers and their modified version in form of an optical klystron. We then provide a short facility review of the OK-4/Duke SR FEL, as both simulations and experiments were carried out at this facility. We then report on the giant pulse mode of operation and present both experiment al and numerical data on this mode of operation. The evolution of the e-beam was studied in giant pulse mode and results compared to experimental data captured on streak camera and reconstructed by phase-space tomography. The results of the comparison showed very good agreement between experimental data and simulated data. Giant pulse mode makes it possible to obtain GW levels of intra-cavity peak power and this made it feasible to carry out harmonic generation experimen ts. Results from proof-of -principle experiments carried out at OK-4/Duke SR FEL which resulted in coherent harmonic generation down to wavelengths of \begin_inset Formula $37$ \end_inset \begin_inset Formula $nm$ \end_inset is reported. \layout Chapter Conclusions And Future Work \layout Standard The interest kindled by the scientific potential of X-ray free electron lasers provide motivation for achieving higher brightness in electron beams. Photo-injectors have enjoyed considerable success to serve as sources for these high brightness electron beams. We have presented in this thesis efforts to integrate digital light processing with present day photo-injector technology to further advance the brightness of electron beams. We report on technique development as well as experimental and numerical studies undertaken to enhance our understanding of the physics and technology of high brightness electron beams. \layout Standard We have developed a new scheme to measure the quantum efficiency with higher spatial resolution of a photo-cathode. This new found grasp over higher spatial resolution of the quantum efficiency is significant as the uniformity of the quantum efficiency has an impact on the uniformity of the electron beam and consequently on its brightness. The present implementation of this technique also lays the ground work for its future use as a corrective device. Such a device will be able to compensate for the non-uniformity in quantum efficiency of the photo-cathode surface by illuminating different sections of the cathode with different intensity of laser light, thus generating uniform density of electrons at the cathode. \layout Standard We report on experimental studies undertaken at Accelerator Test Facility at Brookhaven National laboratory , NY, to comprehend the effect of distortions arising in the drive laser beam on the emittance and thus on brightness of electron beams. We find that macroscopic non-uniformities in the laser beam distribution adversely affect beam quality though the effect is not as pronounced in the presence of fine grained distortions in the laser beam distribution. These experiments were carried out by utilizing a commercial and relatively inexpensive digital light processing kit as a spatial light modulator. The use of spatial light modulators to hunt for the "ideal" distribution of the laser beam is an active field of research in the high brightness electron beam community. Concurrent research is ongoing to fabricate and use specialised mirrors, phase plates and liquid crystals to modulate the transverse distribution of the drive laser. Though all these spatial light modulators have relative merits, the ease of availability and standardisation of the digital light processing technique perhaps makes it a leading candidate in this area of research. The present research in this area focuses on active laser beam shaping done by numerical codes (genetic or annealing algorithms) that communicate with the digital light processing unit and optimise a user defined global parameter. It has been shown in some recent research \begin_inset LatexCommand \cite{key-73} \end_inset that emittance maybe used as such a global parameter in order to improve brightness of electron beams. Once again the study presented in this thesis, reporting on the first use of DMD for high brightness photo-injectors, prepares ground for future use of the light processing technology to actively shape laser beams. \layout Standard The present studies were limited by resolution of existing electron beam diagnostics and by the performance repeatability of the ATF linear accelerator. Future development of the technique requires better and more sensitive diagnostics to realize the vision of shaping electron beams to achieve ultimate brightness. \layout Chapter \paragraph_spacing double \start_of_appendix Quadrupole as Focusing element \begin_inset LatexCommand \label{cha:AppendixAQuadrupole-as-Focusing} \end_inset \layout Standard Magnetic focusing schemes in accelerators often employ quadrupoles. We derive the fields obtained in quadrapoles and discuss there focusing properties. We derive the electric or magnetic fields from the scalar potential \begin_inset Formula $\psi$ \end_inset obeys two dimensional Laplace equation in the limit that the longitudinal dimension of the device is much larger than its transverse dimensions. In cylindrical coordinates of \begin_inset Formula $(\rho,\phi,z)$ \end_inset \begin_inset Formula \begin{equation} \nabla_{\perp}\psi=0\label{eq:2DLaplcae}\end{equation} \end_inset \layout Standard the solutions to this equation are of the form \begin_inset Formula \begin{equation} \psi=\sum_{n=1}^{\infty}a_{n}\rho^{n}\cos(n\phi)+b_{n}\sin(n\phi)\label{eq:AppQuad_Soln2DLaplace}\end{equation} \end_inset \layout Standard For n=1 \begin_inset Formula \begin{equation} \psi_{1}=a\rho\cos(\phi)+b\rho sin(\phi)=a\hat{x}+b_{1}\hat{y}\label{eq:AppQuad_FirstTerm}\end{equation} \end_inset \layout Standard For magnetic field \begin_inset Formula \begin{equation} \vec{B_{1}}=-\nabla\psi_{1}=--a_{1}\hat{x}-b_{1}\hat{y}\label{eq:AppQuad_DipoleTerm}\end{equation} \end_inset \layout Standard Equation \begin_inset LatexCommand \ref{eq:AppQuad_DipoleTerm} \end_inset , is referred to as dipole field as such a field can be formed with a magnet possessing two poles. \layout Standard For n=2 \begin_inset Formula \begin{equation} \psi_{2}=a^{2}\rho\cos(2\phi)+b^{2}\rho sin(2\phi)=a_{2}(x^{2}-y^{2})+2b_{2}xy\label{eq:AppQuad_SecondTerm}\end{equation} \end_inset \layout Standard The associated magnetic field is \begin_inset Formula \begin{equation} \vec{B_{2}}=2a_{2}(-x\hat{x}+y\hat{y})-2b_{2}(y\hat{x}+x\hat{y})\label{eq:AppQuad_QuadrapuleTerm}\end{equation} \end_inset \layout Standard If the term a \begin_inset Formula $_{\textrm{2}}$ \end_inset is non vanishing then the force in x dimension is proportional to magnitude in y and vice versa. Such a quadrapole couples the x and y motion and is usually undesirable for accelerator applications.Normal quadrupole=poles are designed to have vanishing b \begin_inset Formula $_{\textrm{2}}$ \end_inset and non vanishing a \begin_inset Formula $_{\textrm{2}}$ \end_inset .The force can thus be written as \begin_inset Formula \begin{equation} F_{\perp}=qv\hat{z}\times\vec{B_{2}}=qvb_{2}(y\hat{y}-x\hat{x})\label{eq:AppQuad_Force}\end{equation} \end_inset \layout Standard Assuming qv \begin_inset Formula $_{\textrm{z}}$ \end_inset is positive then the force is focusing in x and defocussing in y. For eV \begin_inset Formula $_{\textrm{z}}$ \end_inset is negative the force is focusing in y and defocussing in y. The strength of this field is given by \begin_inset Formula \begin{equation} b_{2}=-\frac{\partial_{x}B_{x}}{\partial y}\equiv\frac{B'}{2}\label{eq:AppQuad_fieldstrength}\end{equation} \end_inset \layout Standard Under the paraxial approximation near z axis, we can write transverse equations of motion as \begin_inset Formula \begin{equation} F_{x}=\gamma m_{o}v_{o}x''=\frac{-qB'}{p_{o}}x\label{eq:AppQuad_ForceX}\end{equation} \end_inset \layout Standard and \begin_inset Formula \begin{equation} F_{y}=\gamma m_{o}v_{o}y''=\frac{-qB'}{p_{o}}y\label{eq:AppQuad_ForceY}\end{equation} \end_inset \layout Standard The equations \begin_inset LatexCommand \ref{eq:AppQuad_ForceX} \end_inset and \begin_inset LatexCommand \ref{eq:AppQuad_ForceY} \end_inset can be written as \begin_inset Formula \begin{eqnarray} x'' & +\kappa_{o}^{2}x & =0\\ y'' & -\kappa_{o}^{2}y & =0\end{eqnarray} \end_inset \layout Chapter Digital Mirror Device \begin_inset LatexCommand \label{dmd} \end_inset \layout Standard \paragraph_spacing double The DMD is an array of aluminium micro-mirrors monolithically fabricated over an array of CMOS \begin_inset LatexCommand \label{Appendix_CMOS} \end_inset SRAM \begin_inset Foot collapsed false \layout Standard complementary metal oxide semiconductor with static random access memory \end_inset cells.Each SRAM cell corresponds to a micro-mirror and allows each mirror to be individually addressed so as to rotate \begin_inset Formula $\pm12^{o}$ \end_inset ,limited by mechanical stops.This is made possible by creating air gaps between the metal layers of the super structure,making it possible for the structure to rotate.The mirror is connected to an underlying yoke which in turn is suspended by two thin torsion hinges to support posts.This micro-mirror superstructure is fabricated through a series of aluminium metal depositions,ox ide masks,metal etches,and organic spacers \begin_inset LatexCommand \cite{key-28} \end_inset . Fig shows an exploded view of the DMD. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/dmd_2pixelANDdmd_exploded.eps \end_inset \layout Caption Exploded view of a DMD pixel. \layout Standard Picture courtesy Texas Instruments \end_inset \layout Standard \paragraph_spacing double The mirror is rotated as a result of electrostatic attraction between the mirror structure and the underlying memory cell.The mirror and the yoke are connected to a bias/reset voltage.The address electrodes are connected to the underlying CMOS memory through via contacts.The mirror moves as a 1 or 0 is stored in the memory cell,that is one address electrode is at ground and the other at some potential \begin_inset Formula $V_{DD}$ \end_inset .This applies a bias voltage to the mirror/yoke structure and the mirror is attracted to the side with the largest electrostatic field differential as shown in fig.To release the mirror,a short reset pulse is sent and the bias voltage is removed.Each micro-mirror is 16 \begin_inset Formula $\mu m^{2}$ \end_inset .The array places each micro-mirror on a 17 \begin_inset Formula $\mu m$ \end_inset pitch leaving a gap of less than \begin_inset Formula $1\mu m$ \end_inset between the micro-mirrors. \layout Chapter \paragraph_spacing double Server Application \layout Standard \paragraph_spacing double The server is a multi threaded network based windows application that simultaneo usly controls the digital mirror device and is receptive to inputs from the client application.This application is written using the MFC API of Visual C++ with a dialog class and has windows sockets class incorporated into it.The windows socket class defines a network programming interface designed for Microsoft windows.In this application we follow the TCP/IP protocol for sockets. The socket class \emph on CMySocketServer \emph default is created with Csocket as the base class and CMySocketServer.cpp is the implementation file. \layout Standard \paragraph_spacing double The main implementation file for the server application is D100DemoDlg.cpp and below is a list of the critical member functions along with excerpts of the code.The success of any socket function is reported by 1. \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Socket2.eps \end_inset \end_inset \layout Standard \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/Socket1.eps lyxscale 50 clip \end_inset \layout Caption Flowchart depicting the socket functions of the server program \end_inset \layout Section OnBListen(Activated by button listen) \layout Standard This creates a socket m_sListen Socket and ensures listening on port 4000. \layout LyX-Code int nret; \layout LyX-Code strServName=m_strServName; //place in local variable \layout LyX-Code int iErr1=m_sConnectSocket.Create(); //create a socket \layout LyX-Code nret=m_sConnectSocket.Connect(strServName,m_iPort); \newline //connect to given port and server name \layout LyX-Code printf(" \backslash n Success should return 1 fo nret"); \layout LyX-Code printf(" \backslash n Connect Socket returns %i",nret); \layout Section OnAccept(Not user activated) \layout Standard Accepts connections from a remote client as well as receives the files from a connetced client. \layout Subsection Accepts socket connection \layout LyX-Code struct sockaddr_in inservice; \layout LyX-Code int sz=sizeof(struct sockaddr_in); \layout LyX-Code if(m_sListenSocket.Accept(m_sConnectSocket, \newline (LPSOCKADDR)&inservice,&sz)==INVALID_SOCKET) \layout LyX-Code { \layout LyX-Code //AfxMessageBox("ACCEPT failed"); \layout LyX-Code CString message =" \backslash n Accept failed due to invalid socket \backslash n "; \layout LyX-Code LogMessage(message); \layout LyX-Code m_sConnectSocket.Close(); \layout LyX-Code } \layout Subsection Receives file \layout Standard The server first receoves the size of the incoming file in bytes nd hence knows the expected number of bytes. \newline It then runs in a loop creating a temporary buffer and writing the buffer into a file,and the process continues \newline till all the expected bytes have reached the server. \layout LyX-Code recdData=new BYTE[RECV_BUFFER_SIZE+32]; \layout LyX-Code cbLeftToReceive=dataLength; \newline //cbLeftToReceive is //number to be received \layout LyX-Code do \layout LyX-Code { \layout LyX-Code printf(" \backslash n entering loop to receive the file \backslash n"); \layout LyX-Code int iGet,iRecd; \layout LyX-Code iGet =(cbLeftToReceive0); //do loop continues \newline ends when cbLeftToReceive=0 \layout Chapter Client Application \layout Standard \paragraph_spacing double The client application is a windows based socket application which is created primarily for remote accelerator users to be able to communicate and control the DMD over the internet.It is an interactive GUI enabled application which allows the remote users to send images over the network to the device as well as set the mirrors of the DMD to reset(float) condition. The main features of the client code are listed below along with a brief description of the involved features. \layout Section OnBConnect(activated by Connect button) \layout Standard \paragraph_spacing double \align left This creates a socket called m_sConnectSocket and then establishes a connection with the IP address specified by the \newline user in the interactive GUI.If the connection fails it returns as error value and ensures that a clean destruction of \newline the socket m_sConnectSocket occurs.The relevant part of the code is reproduced below \layout LyX-Code \paragraph_spacing double \layout LyX-Code CString strServName; \newline //local variable to store the server name string \layout LyX-Code int nret; \layout LyX-Code strServName=m_strServName; //place in local variable \layout LyX-Code int iErr1=m_sConnectSocket.Create(); //create a socket \layout LyX-Code nret=m_sConnectSocket.Connect(strServName,m_iPort); \newline //connect to given port and server name \layout LyX-Code printf(" \backslash n Success should return 1 fo nret"); \layout LyX-Code printf(" \backslash n Connect Socket returns %i",nret); \layout Section OnFileOpen(Activated by Browse button) \layout Standard This allows the user to choose an image file located any where on the hard disk of the computer running the client application.This section of the code not only the browsing and loading of an image file,it also converts it into an image that is acceptable by the DMD, relieving the server of the image processing and thus ensuring a very quick upload of images at the rate of 6-10 Hz compared to the 1Hz loading time taken by the acticeX control. \layout LyX-Code //Transform the image from its original bits per pixel \layout LyX-Code //representation to a 1 bit per pixel \newline representation and store it in \layout LyX-Code //the blocks buffer for loading on the device. \layout LyX-Code \layout LyX-Code //int X, Y; \layout LyX-Code COLORREF pixCol, cutOff; \layout LyX-Code cutOff = RGB(240, 240, 240); \layout LyX-Code //Any COLORREF value below cutOff will be turned to black, \layout LyX-Code \newline // any thing above will be white \newline \newline for(int rw = 0; rw < 768; rw++) \newline { \layout LyX-Code for(int bitCnt = 15; bitCnt > -1; bitCnt--) \newline { \newline for (int cl = bitCnt; cl < 1024; cl+=16) \newline { \newline pixCol = GetPixel(hdcSrc, cl, rw ); \newline if (pixCol > cutOff) \newline Dest[ByteCnt] |= (BYTE)(1<0);//loop continues to send \newline //all bytes in the chunk \layout LyX-Code \newline //of size SEND_BUFFER_SIZE \newline printf(" \backslash n outside the sendThisTime loop %i", \newline while(cbLeftToSend>0);//loop continues till all bytes \layout LyX-Code in the file are sent \newline printf(" \backslash n loop for sending entire file over \backslash n"); \layout Section OnBClose(Activated by the Close button) \layout Standard This closes the connection to the sever and ensures that the client is once again put in interactive mode for future connections. \layout Chapter Doppler Effect \begin_inset LatexCommand \label{doppler} \end_inset \layout Standard \paragraph_spacing double \begin_inset Float figure wide false collapsed false \layout Standard \begin_inset Graphics filename C:/Documents and Settings/Samadrita/My Documents/My Pictures/doppler.eps \end_inset \end_inset The Doppler effect is observed with visible light and all other electromagnetic waves. Just as in the case of sound waves, the wavelength in the direction of the source motion is shortened to \begin_inset Formula \[ \lambda"=(c-v_{s})T\] \end_inset \layout Standard where \begin_inset Formula $T_{o}$ \end_inset is the period in the source frame. \layout Standard where all quantities here are measured in the observer's frame. To relate this to the source frequency, it must be expressed in terms of by using the time dilation expression \layout Standard \begin_inset Formula \[ \lambda"=\frac{(c-v_{s})T_{o}}{\sqrt{1-\frac{v_{s}^{2}}{c^{2}}}}\] \end_inset \layout Standard If we define a receding velocity as positive then \layout Standard \begin_inset Formula \[ \lambda"=\frac{(c+v_{s})T_{o}}{\sqrt{1-\frac{v_{s}^{2}}{c^{2}}}}=\gamma(c+v_{s})\frac{\lambda_{o}}{c}\] \end_inset \layout Standard \begin_inset Formula \[ \lambda"=\gamma(1+v_{s}/c)\lambda=\gamma(1+\beta)\lambda\] \end_inset \layout Standard For \begin_inset Formula $\beta\sim1$ \end_inset the abobe equation becomes \layout Standard \begin_inset Formula \[ \lambda"=2\gamma\lambda\] \end_inset \layout Standard \begin_inset ERT status Open \layout Standard \backslash backmatter \end_inset \layout Bibliography \bibitem {key-83} Duncan Graham Rowe, 24 October 2001, "Electron Beams could be used to irradiate post", New Scientist, 12:20 (Print edition). \layout Bibliography \bibitem {key-61} Kendrew JC et al, "A Three-Dimensional Model of the Myoglobin Molecule Obtained by X-ray Analysis", Nature (Mar 8) 181(4610):662-6. \layout Bibliography \bibitem {key-62} Landau and Lifshitz, "The Classical Theory of Fields", Vol2, p. 5. \layout Bibliography \bibitem {key-64} J.Smedley et al, May 2003, "Emittance Measurement with a Pulsed Power Photo-injec tor", Proc. PAC'03, Portland. \layout Bibliography \bibitem {key-63} TOR O.Raubenheimer, November 1994, "Electron Beam Acceleration and Compression for Short Wavelength FELs", SLAC-PUB-6709. \layout Bibliography \bibitem {key-22} J.D.Lawson, 1988, "The Physics of Charged-Particle beams", \begin_inset Formula $2^{nd}$ \end_inset edition, Clarendon Press, Oxford, p. 210. \layout Bibliography \bibitem {key-67} S.G.Anderson et al, 2002, "Space-charge Effects in High Brightness Electron Beam Emittance Measurements", Physical Review Special Topics -Acc. and Beams, Vol 5, p.014201. \layout Bibliography \bibitem {key-23} K.J.Kim, 1989, "RF and Space-Chage Effects In Laser-Driven RF Electron Guns", Nucl. Instrum. and. Meth.A 275 p. 201. \layout Bibliography \bibitem {key-68} S.Schreiber, July 2002, "Performance of TTF Photo-Injector for FEL Operation \begin_inset Quotes erd \end_inset , Proc. of the Workshop of The Physics and Applications of High Brightness Electron Beams,Chia Laguna, Sardinia. \layout Bibliography \bibitem {key-70} S.M. Gurner & M.Tigner, eds, 4 July 2001, "Study for Proposed Phase I ERL Synchrotron Light Source at Cornell University", JLAB -ACT-0-04. \layout Bibliography \bibitem {key-71} Alpha-X Project, http://phys.strath.ac.uk/alpha-x. \layout Bibliography \bibitem {key-69} K-J Kim, May 2003, "High Brightness Electron Beam For X-ray FELS", COOL03, Mt.Fuji, Japan. \layout Bibliography \bibitem {key-74} J.Rosenzweig, "Fundamentals of Beam Physics", Oxford University Press, p. 121. \layout Bibliography \bibitem {key-73} H.Tomizawa et al, "Status of SPRING-8 Photocathode RF Gun for Future Light Sources", private communication. \layout Bibliography \bibitem {key-24} J.Rosenweig and E.Colby, 1995, in "Advanced Accelerator Concepts", ed. P.Schoessow, AIP Conference Proc.335, p. 724. \layout Bibliography \bibitem {key-75} F.Zhou et al, 2003, "Experimental Characterisation of Emittance Growth Induced by Nonuniform Transverse Emittance", Physical Review Special Topics, Vol5, p.094203. \layout Bibliography \bibitem {key-76} M.Reiser, 1994, "Theory and Design Of Charged Particle Beams", John Wiley and Sons Inc, p. 471. \layout Bibliography \bibitem {key-25} E.P.WagnerII et al, 1995, "Construction and Evaluation of a visible spectrometer using Digital Micrommirror Spatial Light Modulation", Applied Spectroscopy, 49, p. 1715 . \layout Bibliography \bibitem {key-26} M.Liang et al, 1997, "Confocal Pattern Period in Multiple -Aperture Confocal Imaging Systems with Coherent Illumination", Opt.Lett.22, p. 751-753. \layout Bibliography \bibitem {key-27} http://accelconf.web.cern.ch/accelconf/pac97/papers/pdf/3W020.PD. \layout Bibliography \bibitem {key-28} L.J.Hornbeck, \begin_inset Quotes erd \end_inset Digital Light Processing and MEMS: Timely Convergence for a Bright Future \begin_inset Quotes erd \end_inset (invited plenary paper), SPIE, Vol 2639, p. 2, Abstract Only. \layout Bibliography \bibitem {key-29} http://www.dlp.com/dlp_technology/ images/dynamic/white_papers/ \newline 154_Hinge_Memory_Paper_IRPS2002.pdf \layout Bibliography \bibitem {key-30} O.Svelto, 1988, "Principles of Lasers", Plenum Press, NewYork. \layout Bibliography \bibitem {key-66} L.Serafini et al, June 1997, "Envelope Analysis of Intense Relativistic Quasilami nar Beams in RF Photoinjectors", Physical Review E, Vol 55, Number 6 p.7565. \layout Bibliography \bibitem {key-31} V.N. Litvinenko et al, 1999, "OK-4/Duke Storage Ring FEL Lasing in the Deep UV", Nucl.Instr. and Meth A 429, p. 151. \layout Bibliography \bibitem {key-32} T.Hara et al, 1994, "Observations of the Super-ACO FEL micropulse with Streak Camera", Nucl. Instr. and Meth.A 341 p. 21. \layout Bibliography \bibitem {key-33} I.V.Pinayev et al, 2001, "Giant high peak power pulses in UV OK-4/Duke Storage Ring FEL Using the Gain Modulator", Nucl. Instr. and Methods A, 475, p. 222. \layout Bibliography \bibitem {key-34} V.N.Litvinenko et al, 1996, "Duke Storage Ring UV/VUV FEL: Status and Prospects \begin_inset Quotes erd \end_inset , Nucl. Instr. and Meth. A. 375 p. 46. \layout Bibliography \bibitem {key-35} V.N.Litvinenko, Jan22-25, 1996, "X Ray Storage Ring FEL: New Concepts and Directions", in Proc. of the 10th ICFA Beam Dynamics Panel Workshop 4th generation light sources Grenoble, France, p. WG6-16. \layout Bibliography \bibitem {key-36} V.N.Litvineko et al, 1995, "Giant Laser Pulses in the Duke Storage Ring UV FEL", Nucl. Instr. and Meth. A 358 p. 334. \layout Bibliography \bibitem {key-37} J.D Jackson, 1975, "Classical Electrodynamics", John Wiley and Sons,. \layout Bibliography \bibitem {key-39} V.N.Litvinenko et al, 1995, "Dynamics of the Duke Storage Ring UV FEL", Nucl. Instr. and Meth. A, 358, p. 369\SpecialChar \@. \layout Bibliography \bibitem {key-41} V.N.Litvinenko, 2003, "New Results and Prospects for Harmonic Generation in Storage Ring FELs", Nucl. Instr. and Meth. A, 507, p. 265\SpecialChar \@. \layout Bibliography \bibitem {key-44} C. Brau, "Free Electron Lasers", Academic Press, INC, NY, p. 83. \layout Bibliography \bibitem {key-47} C. Brau,Free Electron Lasers,Academic Press,INC, NY, p. 66 \layout Bibliography \bibitem {key-49} Srinivas Krishnagopal et al, October 2004, "Free Electron Lasers", Current Science, Vol 87, No.8, p. 25. \layout Bibliography \bibitem {key-50} X-ray data booklet, January 2001, LBNL/PUB-490 Rev2, Chapter2, Berkley. \layout Bibliography \bibitem {key-52} V.N.Litvinenko, "Physics of Super-Pulses in Storage Ring Free Electron Lasers", C-A/AP#181(http://www.agsrhichome.bnl.gov/AP/ap_notes/ap_note_181.pdf) \layout Bibliography \bibitem {key-53} E.C.Longhi, "Coherent Harmonic Generation in Storage Ring Free Electron Lasers \begin_inset Quotes erd \end_inset , Ph.D. Dissertation, Physics Department, Duke University (to be completed). \layout Bibliography \bibitem {key-54} K.Chalut, 2005, "New Methods in Phase-Space Tomography and its Application to Electron Beam Dynamics in Storage Rings", Ph.D Dissertation, Physics Department, Duke University. \layout Standard \begin_inset ERT status Open \layout Standard \backslash newpage \layout Standard \backslash chapter{BIOGRAPHY} \layout Standard Samadrita Roychowdhury was born to Anjulika and Dhruvjyoti Roychowdhury in Lucknow, India on January 22nd, 1976. Samadrita attended schools in various cities of India, moving often as a consequence of her father's federal job and graduated from high school in 1994 from St. Joseph's Convent Jabalpur. \layout Standard She obtained her bachelors in science from Sri Sathya Sai college, Bhopal, in 1997. She continued her education at the Indian Institute of Technology, Madras, India and was awarded a degree of M.Sc in physics in 1999. Her thesis work at the Indian Institute of technology (I.I.T. Madras) was in two dimensional chaos theory and was supervised by Dr. Neelima M. Gupte. She pursued her interest in non linear dynamics as a research scholar at the Indian Institute of Science, Bangalore, for a year, before joining Duke University, department of physics in the year 2000 for a doctoral degree. \layout Standard Samadrita continued to work in non linear dynamics in the summer and fall of 2001, before switching to Free Electron Laser physics in the spring semester of 2002, joining the group headed by Prof.V.N. Litvinenko. Working on free electron laser physics,she was awarded a Master of Arts in physics in 2003 by Duke University. She went on to complete her doctoral dissertation working at and collaborating with the Accelerator Test Facility, at Brookhaven National Laboratory, NY. She expects to receive her Doctor of Philosophy in Physics in May 2006 from Duke University. \layout Standard \end_inset \the_end