Table of Contents

Preface XIII


1 Beams of Electrons or Photons 1


1.1 Preview 1


1.2 Coordinate Definitions 2


1.3 One-dimensional Transverse Propagation Equations 4


1.4 Transfer Matrices for Simple Elements 7


1.4.1 Drift Space 7


1.4.2 Thin Lens 7


1.4.3 Thick Lens 9


1.4.4 Erect Quadrupole Lens 11


1.5 Elliptical (in Phase Space) Beams 13


1.6 Beam Envelope E(s) 15


1.7 Gaussian Beams: their Variances and Covariances 17


1.8 Pseudoharmonic Trajectory Description 18


1.9 Transfer Matrix Parametrization 21


1.10 Reconciliation of Beam and Lattice Parameters 23


1.10.1 Beam Evolution Through a Drift Section 23


1.10.2 Beam Evolution Through a Thin Lens 24


References 25


2 Beams Treated as Waves 27


2.1 Preview 27


2.2 Scalar Wave Equation 28


2.3 The Short Wavelength, Geometric Optics Limit 30


2.3.1 Determination of Rays from Wavefronts 31


2.3.2 The Ray Equation in Geometric Optics 32


2.3.3 Obtaining Phase Information from Intensity Measurement 34


2.4 Wave Description of Gaussian Beams 36


2.4.1 Gaussian Beam in a Focusing Medium 37


2.4.2 Spatial Dependence of a Wave Near a Free Space Focus 39


2.4.3 The ABCD Law 40


2.4.4 Optics Using Mirrors 42


2.4.5 Wave Particle Duality for Electrons 44


2.5 Synchrotron Radiation: Waves or Particles? 45


2.6 X-ray Holography and Phase Contrast and Lens-free Imaging 46


References 48


3 Synchrotron Radiation From Accelerator Magnets 49


3.1 Capsule History of Synchrotron Light Sources 49


3.2 Generalities 51


3.3 Potentials and Fields 53


3.4 Relations Between Observation Time and Retarded Time 54


3.5 Evaluation of Electric and Magnetic Fields 58


3.5.1 Radial Field Approximation 60


3.6 Total Power Radiated and its Angular Distribution 62


3.7 Spectral Power Density of the Radiation 65


3.7.1 Estimate of Frequency Spectrum from Pulse Duration 65


3.7.2 Radial Approximation 67


3.7.3 Accurate Formula for Spectral Power Density 68


3.8 Radiation from Multiple Charges 70


3.9 The Terminology of “Intensity” Measures 71


3.10 Photon Beam Features “Inherited from” the Electron Beam 74


3.11 Intensity Estimates for Bending Magnet beams 75


References 81


4 Simple Storage Rings 83


4.1 Preview 83


4.2 The Uniform Field Ring 84


4.3 Horizontal Stability 85


4.4 Vertical Stability 87


4.5 Simultaneous Horizontal and Vertical Stability 88


4.6 Dispersion 89


4.7 Momentum Compaction 90


4.8 Chromaticity 91


4.9 Strong Focusing 92


5 The Influence of Synchrotron Radiation on a Storage Ring 95


5.1 Preview 95


5.2 Statistical Properties of Synchrotron Radiation 96


5.2.1 Total Energy Radiated 96


5.2.2 The Distribution of Photon Energies; “Regularized Treatment” 99


5.2.3 Randomness of the Radiation 103


5.3 The Damping Rate Sum Rule: Robinson’s Theorem 105


5.3.1 Vertical Damping 110


5.3.2 Longitudinal Damping 111


5.3.3 Horizontal Damping and Partition Numbers 119


5.4 Equilibrium between Damping and Fluctuation. 120


5.5 Horizontal Equilibrium and Beam Width 121


5.6 Longitudinal Bunch Distributions 128


5.6.1 Energy Spread 128


5.6.2 Bunch Length 131


5.7 “Thermodynamics” of Wiggler-dominated Storage Rings 132


5.7.1 Emittance of Pure Wiggler Lattice 132


5.7.2 Thermodynamic Analogy 137


References 140


6 Elementary Theory Of Linacs 141


6.1 Acknowledgement and Preview 141


6.2 The Nonrelativistic Linac 142


6.2.1 Transit Time Factor 142


6.2.2 Shunt Impedance 143


6.2.3 Cavity Q, R/Q, and Decay Time 144


6.2.4 Phase Stability and Adiabatic Damping 145


6.2.5 Transverse Defocusing 151


6.3 The Relativistic Electron Linac 153


6.3.1 Introduction 153


6.3.2 Particle Acceleration by a Wave 154


6.3.3 Wave Confined by Parallel Planes 155


6.3.4 Circular Waveguide 161


6.3.5 Cylindrical “Pill-box” Resonator 163


6.3.6 Lumped Constant Model for One Cavity Resonance 164


6.3.7 Cavity Excitation 166


6.3.8 Wave Propagation in Coupled Resonator Chain 168


6.3.9 Periodically Loaded Structures 172


6.3.10 Space Harmonics 174


7 Undulator Radiation 179


7.1 Preview 179


7.2 Introduction 180


7.3 Electron Orbit in a Wiggler or Undulator 184


7.4 Energy Radiated From one Wiggler Pole 187


7.5 Spectral Analysis for Arbitrary Longitudinal Field Profile 188


7.6 Spectrum of the Radiation from a Single Pole 191


7.6.1 Orbit Treated as Arc of Circle 191


7.6.2 Radiation from a Single, Short, Isolated, Magnet, K _ 1 192


7.7 Coherence from Multiple Deflections 196


7.8 Phasor Summation for K _ 1 199


7.9 Photon Energy Distributions 203


7.9.1 Energy Distribution from the n = 1 Undulator Fundamental 203


7.10 Undulator Radiation for Arbitrary K Value 204


7.10.1 Analytic Formulation 204


7.10.2 Diffraction Grating Analogy 212


7.10.3 Numerical/Graphical Representation of Undulator Radiation 213


7.10.4 Approximation of the Integrals by Special Functions 220


7.10.5 Practical Evaluation of the Series 222


7.11 Post-monochromator Profile 223


7.11.1 Monochromatic Annular Rings 225


7.11.2 Numerical Investigation of Undulator Rings 227


7.11.3 Is the Forward Undulator Peak Subject to Angular Narrowing? 227


References 230


8 Undulator Magnets 231


8.1 Preview 231


8.2 Considerations Governing Undulator Parameters 232


8.3 Simplified Radiation Formulas 234


8.4 A Hybrid, Electo-permanent, Asymmetric Undulator 237


8.4.1 Electromagnet Design 239


8.4.2 Permanent Magnet Design—Small Gap Limit 244


8.4.3 Combined Electro-/Permanent- magnet Design 246


8.4.4 Estimated X-ray Flux 247


References 248


9 X-Ray Beam Line Design 249


9.1 Preview 249


9.2 Beam Line Generalities 250


9.3 Accelerator Parameters 252


9.4 Bragg Scattering and Darwin Width 254


9.5 Aperture-defined Beam Line Design 257


9.5.1 Undulator Radiation, n = 1, Negligible Electron Divergence 257


9.5.2 Effect of Electron Beam Emittances on Flux and Brilliance 261


9.5.3 Brilliance with K > 1 and n > 1 263


9.6 X-ray Mirrors 266


9.6.1 Specular Reflection of X-rays 266


9.6.2 Elliptical Mirrors 266


9.6.3 Hyperbolic Mirrors 267


9.7 X-ray Lenses 270


9.7.1 Monochromatic X-ray Lens 270


9.7.2 Focusing a Monochromatic Undulator Radiation Ring 270


9.7.3 Undulator-specific X-ray Lens 272


9.7.4 Lens Quality 278


9.8 Beam cameras 279


9.8.1 The Pin-hole Camera 279


9.8.2 Imaging the Beam with Visible Light 280


9.8.3 Practicality of Lens-based, X-ray Beam Camera? 283


9.9 Aperture-free X-ray Beam Line Design 286


9.9.1 Aperture-free Rationale 286


9.9.2 Aperture-free Microbeam Line Based on Lenses 287


9.9.3 Effective Lens Stop Caused by Absorption 291


9.9.4 Choice of Undulator Parameter K 292


9.9.5 Estimated Flux 293


9.9.6 Estimated Brilliance and Qualifying Comments 295


References 296


10 The Energy Recovery Linac X-Ray Source 299


10.1 Preview 299


10.2 Introduction 300


10.3 Emittance Evolution in a DC Electron Gun 301


10.4 Qualitative Description of Lattice Design Issues 306


10.5 Isochronous Arc Design 309


10.6 Evolution of Betatron Amplitudes through the Linac Sections 314


10.6.1 Deceleration through Linac Section to Dump 314


10.6.2 Triplet Design 318


10.6.3 Acceleration through the High Energy Linac Section 319


10.7 Emittance Growth Due to CSR and Space Charge 321


References 325


11 A Fourth Generation, Fast Cycling, Conventional Light Source 327


11.1 Preview 327


11.2 Low Emittance Lattices 328


11.3 Production of High Quality X-ray Beams 333


11.3.1 A Formula for Brilliance B 333


11.3.2 A Strategy to Maximize B 334


11.3.3 Hypothetical Utilization of an Existing Large Ring 335


11.4 Acceleration Scenario 338


11.5 Power Considerations 342


11.5.1 Average Power 342


11.5.2 Instantaneous Power 343


11.6 Critical Components and Parameter Dependencies 344


11.7 Trbojevic-Courant Minimum Emittance Cells 346


11.7.1 Basic Formulas 346


11.7.2 Thin Lens Treatment 348


11.7.3 Thick Lens Treatment 353


11.7.4 Zero Dispersion Straight Sections 355


11.7.5 Nonlinearity and Dynamic Aperture 357


11.8 Emittance Evolution During Acceleration 359


11.9 Touschek Lifetime Estimate 362


11.10 Performance as X-ray Source 363


11.10.1 Brilliance from Short Undulator 363


11.10.2 Refinement of Brilliance Calculation 365


References 368


12 Compton Scattered Beams And “Laser Wire” Diagnostics 369


12.1 Preview 369


12.2 Compton Scattering Kinematics 370


12.3 Some Specialized Laser, Electron Beam Configurations 373


12.3.1 Back-scattered Photons 373


12.3.2 Orthogonal Photon Incidence in the Laboratory 374


12.3.3 Orthogonal Electron Frame Incidence 375


12.4 Total Compton Cross Section 380


12.5 The Photon Beam Treated as an Electromagnetic Wave 381


12.5.1 Determination of the Electron’s Velocity Modulation 382


12.5.2 Undulator Parametrization of Electron Motion in a Wave 386


12.6 Undulator Fields in Electron Rest Frame 388


12.6.1 Some Formulas from Special Relativity [7] 389


12.6.2 Treatment of an Undulator Magnet as an Electromagnetic Wave 389


12.7 Classical Derivation of Thomson Scattering 391


12.7.1 Introduction 391


12.7.2 Free Electron Oscillating in Electromagnetic Wave [5] 392


12.7.3 Electric Dipole Radiation [6] 392


12.7.4 Scattering Rate Expressed as a Total Cross Section 394


12.8 Transformation of Photon Distributions to the Laboratory 395


12.8.1 Solid Angle Transformation 395


12.8.2 Photon Energy Distribution in the Laboratory 396


12.8.3 A Theorem Applicable to Isotropic Distributions 398


12.8.4 Energy Distribution of Undulator Radiation 399


12.9 Rate Estimates for a Laser Wire Diagnostic Apparatus 401


12.9.1 The Nonrelativistic Limit 403


12.9.2 Laser Wire Treated as Undulator 403


12.9.3 Laser Wire Treated via Electron Rest Frame 405


12.9.4 Invariant Cross Section Applied to Laser Wire 406


12.9.5 Bunched Beam Rates 408


References 411


13 Space Charge Effects and Coherent Radiation 413


13.1 Acknowledgement and Preview 413


13.2 Introduction to the String Space Charge Formalism 414


13.3 Self-force of Moving Straight Charged String 418


13.4 Self-force of Moving Straight Charged Ribbon 423


13.5 Curve End Point Determination 430


13.6 Field Calculation 434


13.7 “Regularization” of the Longitudinal Force 438


13.8 Coherent Synchrotron Radiation 440


13.9 Evaluation of Integrals 442


13.10 Calculational Practicalities 443


13.11 Suppression of CSR by Wall Shielding 444


13.12 Effects of Entering and Leaving Magnets 445


13.13 Space Charge Calculations Using Unified Accelerator Libraries 447


13.13.1 Numerical Procedures Used by UAL 448


13.13.2 Program Architecture 450


13.13.3 Numerical Procedures 452


13.13.4 Comparison with TRAFIC4 [24] 453


References 456


14 The X-ray FEL 457


14.1 Absorption and Spontaneous and Stimulated Emission 457


14.2 Closed and Open FELs 458


14.3 Interpretation of Undulator Radiation as Compton Scattering 459


14.4 Applicability Condition for Semi-Classical Treatment 461


14.5 Comparison of Storage Ring, ERL, and FEL 462


References 465


Index 467

About the Author

Richard M. Talman is Professor of Physics at Cornell University, Ithaca, New York. After receiving B.A and M.A. at the University of Western Ontario, he received his Ph.D. at the California Institute of Technology in 1963. Since then he has been at Cornell, accepting a full professorship for Physics in 1971. He has spent terms as visiting scientist at Stanford(2), CERN(2), Berkeley(2) and Saskatchewan, and served as leader of the Instrumentation and Diagnostics Group at the SSC project in Dallas. He has given courses on accelerators at Chicago, Austin, Rice, and Yale. Initially a particle physics experimentalist, Professor Talman has been engaged in the design of a series of accelerators, with recent emphasis on their use for x-ray production.