Cook Thesis

University of California Los Angeles

Generation of Narrow-Band Terahertz Coherent Cherenkov Radiation in a Dielectric Wakefield Structure

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics

by

Alan Matthew Cook

2009

c Copyright by

Alan Matthew Cook 2009

The dissertation of Alan Matthew Cook is approved.

Troy Carter

Chandrashekhar Joshi

Claudio Pellegrini

James Rosenzweig, Committee Chair

University of California, Los Angeles 2009

ii

To Emily, my loving wife, who is very proud of me.

iii

Table of Contents

1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1

1

Electron Beam Physics . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.1

Beam Dynamics Formalism . . . . . . . . . . . . . . . . .

3

1.1.2

Radiation from Electron Beams . . . . . . . . . . . . . . .

7

1.1.2.1

Coherently Radiating Beams . . . . . . . . . . .

9

1.1.2.2

Cherenkov Radiation . . . . . . . . . . . . . . . .

10

1.1.2.3

Transition and Diffraction Radiation . . . . . . .

14

1.2

Wakefield Physics . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.3

Terahertz Radiation

. . . . . . . . . . . . . . . . . . . . . . . . .

18

2 Wakefields in a Cylindrical Dielectric-Lined Waveguide . . . .

20

2.0.1 2.1

Experimental Background . . . . . . . . . . . . . . . . . .

21

Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.1.1

Eigenmodes in a Dielectric-Lined Waveguide . . . . . . . .

25

2.1.1.1

Field Solutions . . . . . . . . . . . . . . . . . . .

27

2.1.1.2

Dispersion Relation . . . . . . . . . . . . . . . . .

32

2.1.1.3

Monopole Modes . . . . . . . . . . . . . . . . . .

36

iv

2.1.1.4

Dipole Modes . . . . . . . . . . . . . . . . . . . .

38

2.1.1.5

Transverse Mode Structure . . . . . . . . . . . .

39

Beam-Driven Wakefields . . . . . . . . . . . . . . . . . . .

44

2.1.2.1

Group Velocity and Pulse Length . . . . . . . . .

46

2.1.2.2

Coherent Excitation of Wakefields

. . . . . . . .

47

Computer Simulation . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.2.1

Case 1: Constant Inner Radius . . . . . . . . . . . . . . .

51

2.2.2

Case 2: Constant Wall Thickness . . . . . . . . . . . . . .

52

2.2.3

Case 3: Varying Bunch Length . . . . . . . . . . . . . . .

52

2.2.4

Beam Energy Loss . . . . . . . . . . . . . . . . . . . . . .

55

Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3 Description of Experiment . . . . . . . . . . . . . . . . . . . . . . .

62

2.1.2

2.2

2.3

3.1

3.2

3.3

The Neptune Accelerator . . . . . . . . . . . . . . . . . . . . . . .

63

3.1.1

Magnetic Chicane Compression . . . . . . . . . . . . . . .

68

Design of the Experiment . . . . . . . . . . . . . . . . . . . . . .

70

3.2.1

Dielectric Structures . . . . . . . . . . . . . . . . . . . . .

71

3.2.2

Experimental Layout . . . . . . . . . . . . . . . . . . . . .

76

3.2.3

Transport of THz Radiation . . . . . . . . . . . . . . . . .

78

3.2.4

Hardware Design . . . . . . . . . . . . . . . . . . . . . . .

86

Measurement Apparatus . . . . . . . . . . . . . . . . . . . . . . .

86

3.3.1

Michelson Interferometer . . . . . . . . . . . . . . . . . . .

87

3.3.2

Radiation Detection . . . . . . . . . . . . . . . . . . . . .

89

v

3.3.2.1

Golay Cell Detector . . . . . . . . . . . . . . . .

90

3.3.2.2

Bolometer Detector . . . . . . . . . . . . . . . . .

91

3.3.2.3

Detector Calibration . . . . . . . . . . . . . . . .

92

Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4 Measurement of Coherent Cherenkov Radiation . . . . . . . . .

94

3.4

4.1

Setup for Measurements . . . . . . . . . . . . . . . . . . . . . . .

95

4.2

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . .

98

4.2.1

Control Measurement . . . . . . . . . . . . . . . . . . . . .

98

4.2.2

Spectral Measurements . . . . . . . . . . . . . . . . . . . .

99

4.2.3

Radiated Energy Measurements . . . . . . . . . . . . . . . 107

4.3

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A Coherent Edge Radiation from a Compact Bend Magnet . . . 114 A.1 Synchrotron and Edge Radiation . . . . . . . . . . . . . . . . . . 114 A.2 CER Measurements at Neptune . . . . . . . . . . . . . . . . . . . 118 A.2.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.2.2 Compressed Electron Bunch Length . . . . . . . . . . . . . 121 B Electromagnetic Wave Equation . . . . . . . . . . . . . . . . . . . 123 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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List of Figures

1.1

Trace space plots. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2

Radiation by a moving charge. . . . . . . . . . . . . . . . . . . . .

8

1.3

Electrons in a bunch radiating coherently. . . . . . . . . . . . . .

9

1.4

Atoms polarized by a passing charge. . . . . . . . . . . . . . . . .

11

1.5

Cherenkov Radiation. . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.6

Coordinate system used in wakefield analysis. . . . . . . . . . . .

16

2.1

Cylindrical coordinate system. . . . . . . . . . . . . . . . . . . . .

26

2.2

Example of graphical solution. . . . . . . . . . . . . . . . . . . . .

37

2.3

Example of HEM1n mode solution. . . . . . . . . . . . . . . . . .

39

2.4

TM0n mode field radial profiles. . . . . . . . . . . . . . . . . . . .

41

2.5

TE0n mode field radial profiles. . . . . . . . . . . . . . . . . . . .

42

2.6

Comparison of TM01 and HEM11 Ez radial profiles. . . . . . . . .

43

2.7

Sz and uEM radial profiles. . . . . . . . . . . . . . . . . . . . . . .

43

2.8

Group velocity and pulse length. . . . . . . . . . . . . . . . . . . .

44

2.9

Total radiated energy vs. σz . . . . . . . . . . . . . . . . . . . . . .

48

2.10 Electric potential difference within driving beam. . . . . . . . . .

50

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2.11 Ez vs. z for increasing outer radius b (oopic). . . . . . . . . . . .

53

2.12 Power spectra for increasing outer radius b (oopic). . . . . . . . .

54

2.13 Scaling of TM0n frequency with b (oopic). . . . . . . . . . . . . .

55

2.14 Ez vs. z for increasing inner radius a (oopic). . . . . . . . . . . .

56

2.15 Decelerating field scaling with a (oopic). . . . . . . . . . . . . . .

57

2.16 Ez vs. z for increasing bunch length σz (oopic). . . . . . . . . . .

58

2.17 Power spectra for increasing σz (oopic). . . . . . . . . . . . . . .

59

2.18 Longitudinal phase space (oopic). . . . . . . . . . . . . . . . . .

60

3.1

Schematic of entire accelerator system. . . . . . . . . . . . . . . .

66

3.2

Photocathode drive-laser system. . . . . . . . . . . . . . . . . . .

67

3.3

Magnetic chicane compression. . . . . . . . . . . . . . . . . . . . .

69

3.4

Measured effect of linac phase on compression. . . . . . . . . . . .

71

3.5

Absorption coefficient and refractive index of fused silica. . . . . .

74

3.6

First version of experimental setup. . . . . . . . . . . . . . . . . .

76

3.7

Second version of experimental setup. . . . . . . . . . . . . . . . .

78

3.8

Photo of experimental setup. . . . . . . . . . . . . . . . . . . . . .

79

3.9

Photo of tube holder installed. . . . . . . . . . . . . . . . . . . . .

80

3.10 Photo of experimental setup. . . . . . . . . . . . . . . . . . . . . .

81

3.11 Circular waveguide elements. . . . . . . . . . . . . . . . . . . . . .

81

3.12 Photos of radiation transport hardware.

. . . . . . . . . . . . . .

83

3.13 Photos of electroforming process. . . . . . . . . . . . . . . . . . .

85

3.14 Multipurpose experimental hardware. . . . . . . . . . . . . . . . .

87

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3.15 Standard Michelson interferometer. . . . . . . . . . . . . . . . . .

89

3.16 Diagram of Golay cell detector. . . . . . . . . . . . . . . . . . . .

91

3.17 Diagram of Si bolometer detector. . . . . . . . . . . . . . . . . . .

92

4.1

Schematic layout of experimental diagnostics. . . . . . . . . . . .

96

4.2

Example of Gaussian autocorrelation. . . . . . . . . . . . . . . . . 100

4.3

Example of sinusoidal autocorrelation. . . . . . . . . . . . . . . . 100

4.4

Raw measured autocorrelation scans for b = 350 µm tube. . . . . 102

4.5

Raw measured autocorrelation scans for b = 400 µm tube. . . . . 103

4.6

Measured power spectra. . . . . . . . . . . . . . . . . . . . . . . . 104

4.7

Measured full bandwidth. . . . . . . . . . . . . . . . . . . . . . . 106

4.8

Side-by-side comparison of spectra. . . . . . . . . . . . . . . . . . 109

4.9

Comparison of THz peak power. . . . . . . . . . . . . . . . . . . . 111

A.1 Edge radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 A.2 An electron in uniform circular motion. . . . . . . . . . . . . . . . 115 A.3 SR angular intensity patterns. . . . . . . . . . . . . . . . . . . . . 116 A.4 ER intensity patterns, finite-edge-length model. . . . . . . . . . . 117 A.5 Setup for CER measurements. . . . . . . . . . . . . . . . . . . . . 118 A.6 CER measurement diagnostics. . . . . . . . . . . . . . . . . . . . 118 A.7 Measured signal vs. beam charge. . . . . . . . . . . . . . . . . . . 119 A.8 CER polarization scans. . . . . . . . . . . . . . . . . . . . . . . . 120 A.9 Measured autocorrelation of CER. . . . . . . . . . . . . . . . . . . 122

ix

List of Tables

2.1

Calculated mode frequencies. . . . . . . . . . . . . . . . . . . . . .

37

2.2

Calculated values of TM01 mode E0 , U, vg , and tpulse . . . . . . . .

47

2.3

Comparison of radiated energy estimates. . . . . . . . . . . . . . .

57

2.4

Summary of predictions in Chapter 2. . . . . . . . . . . . . . . . .

61

3.1

Neptune beam parameters. . . . . . . . . . . . . . . . . . . . . . .

68

3.2

Dielectric tube specifications. . . . . . . . . . . . . . . . . . . . .

72

4.1

Results of measurements. . . . . . . . . . . . . . . . . . . . . . . . 105

4.2

Summary of measured quantities. . . . . . . . . . . . . . . . . . . 110

4.3

Hypothetical case study parameters. . . . . . . . . . . . . . . . . 112

x

Acknowledgments

I thank my adviser, Jamie Rosenzweig, for his confidence in me and for the constant optimism that has propelled most of my work over the last five years. He was seldom too quick nor too late in declaring victory. I thank Gil Travish for teaching me a lot about science, writing, job hunting, and appreciating the finer things in life. I thank Sergei Tochitsky for teaching me about laser operation and experimental research, and for always protecting the Neptune team spirit. I thank each person involved in the Particle Beam Physics Laboratory research group; I have enjoyed working with and connecting with each of you in different ways. I also wish to thank the many members of the UCLA Department of Physics and Astronomy staff for their friendship and assistance. I thank the former PBPL students with whom I have worked most closely — Matt Thompson, Pedro Frigola, Gerard Andonian, Pietro Musumeci, Joel England, and Scott Anderson — for the training, guidance, and advice they have given. I am grateful to my closest contemporaries, Mike Dunning, Oliver Williams, Dan Haberberger, and Erik Hemsing, with whom I have shared graduate school, many scientific discussions, and hours of commiseration; I look forward to decades of being colleagues. I thank Harry Lockart and the staff of the UCLA Physical Sciences machine shop, who made most of the hardware used in my experiment, for their skillful work and willingness to go out of their way help me. I thank James Sarrett of Applied Minds, Inc. for providing free EDM work, as well a countable infinity of dinner tabs over the 13 years I’ve known him. I thank RadiaBeam Technologies for use of calibration equipment. I thank the American Institute of Physics

xi

for permission to reprint a figure. I thank Sven Reiche for permission to use his likeness on several tshirt designs. I thank Rodion Tikhoplav and Andrey Knyazik for helping with various aspects of hardware setup for my experiment. I am grateful to Bruce Doak, Ralph Chamberlin, Richard Jacob, and Mike Grams, who were key in my undergraduate education and my decision to continue pursuing physics. I thank Dave Vonk and Kurt Hinterbichler, with whom I shared the Arizona State University undergraduate physics curriculum and whose staggering talent has always inspired and motivated me. I am also grateful to the late John Wheatley, who taught me to use an electron microscope and whose life impacted many at ASU. I thank my parents, Tim and Linda Cook, for supporting me in my education and for the sacrifices they have made for our family. I thank the other members of my family for their encouragement and companionship. Finally, I thank those of my friends and family who have always encouraged me to serve others and to love God: constant and formative struggles.

xii

Vita

1980

Born, Phoenix, Arizona, USA

2000

NASA Space Grant Intern, Arizona State University

2000

Undergraduate Teaching Assistant, Department of Physics and Astronomy, ASU

2001

Research Experience for Undergraduates student, Department of Physics and Astronomy, ASU

2002

REU student, Department of Physics and Astronomy, ASU

2002 – 2003

Undergraduate Research Assistant, Atomic Beam Laboratory, Department of Physics and Astronomy, ASU

2003

John and Richard Jacob Award for Undergraduate Research, Department of Physics and Astronomy, ASU

2003

B.S., Physics, magna cum laude, ASU

2003 – 2004

Teaching Assistant, Department of Physics and Astronomy, University of California, Los Angeles

2004 – 2009

Graduate Student Researcher, Department of Physics and Astronomy, UCLA.

2005

M.S., Physics, UCLA

2005

US Particle Accelerator School, University of California, Berkeley

xiii

2005

US Particle Accelerator School, Cornell University

2006

US Particle Accelerator School, ASU

2008

US Particle Accelerator School, University of California, Santa Cruz

Publications

J.B. Rosenzweig, et al., “Effects of Ion Motion in Intense Beam-Driven Plasma Wakefield Accelerators,” Phys. Rev. Lett. 95, 195002 (2005).

J.B. Rosenzweig, et al., “RF and Magnetic Measurements on the SPARC Photoinjector and Solenoid at UCLA,” Proceedings of the 2005 Particle Accelerator Conference, Knoxville, TN, 2624 (2005).

M. P. Grams, et al., “Microscopic Fused Silica Capillary Nozzles as Supersonic Molecular Beam Sources,” J. Phys. D: Appl. Phys. 39, 930 (2006).

J. B. Rosenzweig, et al., “Emittance Compensation with Dynamically Optimized Photoelectron Beam Profiles,” Nucl. Inst. and Meth. A 557, 87 (2006).

A. M. Cook, et al., “Dielectric Wakefield Accelerating Structure as a Source of Terahertz Coherent Cerenkov Radiation,” Proceedings of the 12th Advanced Accelerator Concepts Workshop, Lake Geneva, WI, 831 (2006).

xiv

J. B. Rosenzweig, et al., “Beam Compression Experiments using the UCLA/ATF Compressor,” Proceedings of the 12th Advanced Accelerator Concepts Workshop, Lake Geneva, WI, 642 (2006).

J. B. Rosenzweig, et al., “Experimental Testing of Dynamically Optimized Photoelectron Beams,” Proceedings of the 12th Advanced Accelerator Concepts Workshop, Lake Geneva, WI, 649 (2006).

M. P. Dunning, et al., “Magnetic Chicane Radiation Studies at the BNL ATF,” Proceedings of the 2006 Free Electron Laser Conferece, Berlin, Germany, 447 (2006).

J. B. Rosenzweig, et al., “Optimum Beam Creation in Photoinjectors using Space-Charge Expansion,” Proceedings of the 2006 Free Electron Laser Conferece, Berlin, Germany, 752 (2006).

A. M. Cook, et al., “Mitigation of RF Gun Breakdown by Removal of Tuning Rods in High Field Regions,” Int. J. Mod. Phys. A 22, 4039 (2007).

G. Andonian, et al., “Observation of Coherent Edge Radiation Emitted by a 100 Femotsecond Compressed Electron Beam,” Int. J. Mod. Phys. A 22, 4101 (2007).

A. M. Cook,et al., “Beam-Driven Dielectric Wakefield Accelerating Structure as a THz Radiation Source,” Proceedings of the 2007 Particle Accelerator Conference, Albuquerque, NM, 3041 (2007).

xv

A. M. Cook, et al., “Mitigation of Electric Breakdown in an RF Photoinjector by Removal of Tuning Rods in High-Field Regions,” Proceedings of the 2007 Particle Accelerator Conference, Albuquerque, NM, 2415 (2007).

G. Travish, et al., “Dielectric Wakefield Accelerator Experiments at the SABER Facility,” Proceedings of the 2007 Particle Accelerator Conference, Albuquerque, NM, 3058 (2007).

D. Alesini, et al., “Experimental Results with the SPARC Emittance-Meter,” Proceedings of the 2007 Particle Accelerator Conference, Albuquerque, NM, 80 (2007).

G. Gatti, et al., “Coherent Cherenkov Radiation as a Temporal Diagnostic for Microbunched Beams,” Proceedings of the 2007 Particle Accelerator Conference, Albuquerque, NM, 998 (2007).

M. C. Thompson, et al., “Breakdown Limits on Gigavolt-per-Meter ElectronBeam-Driven Wakefields in Dielectric Structures,” Phys. Rev. Lett. 100, 214801 (2008).

A. M. Cook, et al., “Status of Coherent Cherenkov Wakefield Experiment at UCLA,” Proceedings of the 13th Advanced Accelerator Concepts Workshop, Santa Cruz, CA (2008).

G. Andonian, et al., “Observation of Coherent Terahertz Edge Radiation from Compressed Electron Beams,” Phys. Rev. ST Accel. Beams 12, 030701 (2009).

xvi

S. Tochitsky et al., “Efficient Harmonic Microbunching in a 7th-Order InverseFree-Electron Laser Interaction,” Phys. Rev. ST Accel. Beams 12, 050703 (2009).

xvii

Abstract of the Dissertation

Generation of Narrow-Band Terahertz Coherent Cherenkov Radiation in a Dielectric Wakefield Structure by

Alan Matthew Cook Doctor of Philosophy in Physics University of California, Los Angeles, 2009 Professor James Rosenzweig, Chair

This study explores the use of a dielectric-lined waveguide structure as a means of producing narrow-band terahertz radiation in the form of electronbeam-driven coherent Cherenkov radiation wakefields. This concept builds on previously studied scenarios such as the Cherenkov maser and the Cherenkov free-electron laser. It is distinct in that it relies solely on coherent wakefield excitation instead of a microbunching instability gain process, in analogy to the superradiant regime of FEL operation. The narrow bandwidth is due to the single-mode nature of the excitation, enabled by the exclusion (due to coherence) of discrete waveguide modes with wavelengths shorter than the driving electron bunch length. This allows an inherently broadband beam current profile to radiate power into a single frequency, which is selectable by appropriate choice of design parameters. The theoretical component of this dissertation is aimed at making predictions for comparison with experimental results. The functional form and propagating

xviii

mode frequencies of the electromagnetic fields in the waveguide structure are found by eigenmode solution in the source-free case beginning from Maxwell’s equations; the response of the structure to a driving electron bunch is then found using a wakefield formalism. Predictions for the frequencies and radiated energy levels obtained from this analysis are corroborated computationally using the commercial particle-in-cell simulation code oopic pro. The experiment is designed to be a proof-of-principle demonstration of the effectiveness of this scenario in converting the energy in an electron beam into electromagnetic radiation. We present detailed measurements showing a narrow emission spectrum peaked at 367 ± 3 GHz from a 1 cm long fused silica capillary tube with sub-mm transverse dimensions, matching the predicted (analytical and computational) TM01 mode resonance to within 1% error. This measurement confirms the expected preferential coherent excitation of the TM01 mode over the HEM11 mode, which lies nearby in frequency but still decisively outside the error estimate established over multiple measurements. The measured 3 dB bandwidth is on the order of . 10% and is seen to be transform-limited. We observe a 100 GHz shift in the emitted central frequency when the tube wall thickness is changed by 50 µm, demonstrating the modular tunability of the source. Calibrated measurements of the radiated energy register up to 10 µJ per 60–80 ps pulse for an incident sub-picosecond electron beam carrying 200 pC of charge, corresponding to a peak power of approximately 150 kW. A case study considering the implementation of this scenario using a 10-cm-long structure with smaller transverse dimensions indicates a possible yield of 50 MW peak power at 1.8 THz and 0.1% bandwidth. This dissertation reports the first direct measurements of narrow-band THz coherent Cherenkov radiation driven by a sub-picosecond electron beam in a die-

xix

lectric wakefield structure, representing a successful adaptation of the previously proven Cherenkov FEL concept to the realm of ultra-short electron beams such as are available in state-of-the-art user facilities around the world. These results prove the potential of this method to produce tunable, narrow-band, pulse-lengthvariable, multi-megawatt peak-power radiation at f > 1 THz in existing modern electron accelerators.

xx

CHAPTER

1

General Introduction

The electron accelerator has emerged as one of the most useful tools for electromagnetic (EM) radiation production in the past decades. Major international light source facilities producing photons at wavelengths from angstroms to millimeters rely on advances in modern high-brightness electron beam physics to enable research spanning the physical and life sciences. One of the current frontiers in radiation source research, at the time of this writing, lies in the region of the EM spectrum between high-frequency microwaves and the far infrared. This range of frequencies, from approximately 100 GHz to 10 terahertz (1 THz = 1012 Hz), is nicknamed the “THz Gap” because of the relatively young state of source technology development, although the designation is quickly becoming outdated. Interest in THz light springs from its wide potential for innovative utility in imaging and spectroscopy. It is no surprise that advanced accelerator research is at the core of many of the most promising avenues in the development of THz sources. Modern stateof-the-art electron beams are capable of delivering electric charge modulated on sub-picosecond (1 ps = 10−12 s) time scales, and thus they are ideally suited for exciting EM fields at ps−1 = THz frequencies. The passage of such a beam

1

through a perturbing element will cause it to radiate readily, and so there are any number of scenarios that one may utilize to convert a beam’s kinetic energy into radiation with properties corresponding to the perturbing structure. One such technique relies on the presence of dielectric material in the vicinity of the beam to produce Cherenkov radiation [1]. The addition of boundaries to the system in the form of conducting material or dielectric interfaces can allow emitted Cherenkov radiation to act back on the driving beam in a process similar to a free-electron laser (FEL) gain mechanism. Alternatively, the boundaries can allow some control over spectral and spatial qualities through mode confinement. This dissertation examines the use of a bounded dielectric structure as a means of converting the energy in a relativistic electron beam with broadband current profile into narrow-band radiation, with a particular view toward the generation of THz radiation. Predictions made through both analytical calculation and computer simulation are discussed and shown to agree very well with the results of an experiment performed at the UCLA Neptune advanced accelerator laboratory. This chapter begins by introducing the physics of electron beams and radiating electrons. This proceeds first in a general sense and then in regard to the Cherenkov radiation process and coherent radiation from beams, which form the fundamental basis for the physical phenomena discussed in the rest of the study. An overview of wakefield physics is then presented, defining a formalism that is useful for making predictions to match our measurements. Finally, a discussion of the applications of THz radiation and current research in THz sources will provide motivation for and underline the importance of the results of this experimental work.

2

1.1

Electron Beam Physics

An electron beam, in a general sense, is a group of electrons collimated into a stream that flows in a vacuum. In practice, a beam can take the form of a continuous stream, a small bunch, or a periodically modulated or microbunched stream. Beams are tailored by a number of means to fit a growing host of applications. Perhaps the foremost application of electron beam technology at the time of this writing is the generation of electromagnetic radiation. As EM waves originate from the disturbance of electric charge, it is natural that they are conveniently produced by a machine offering precise control of a dense packet of charge. The bulk of this section is spent discussing the physics of electron beams in the context of various forms of beam-based radiation.

1.1.1

Beam Dynamics Formalism

There are several extensive and rich treatments of the physics of charged particle beams, such as those in Refs. [2, 3, 4, 5, 6, 7]. Here we offer a basic overview of the terminology and concepts central to a discussion of the dynamics of an electron beam, leaving derivations and specific details to the references. The formalism presented will be referred to in later chapters. The dynamics of a beam containing billions of electrons must be formulated using a collective description, in which the aggregate behavior of the constituent particles is tracked by means of a few variables. The standard collective method describes the behavior of the boundaries of the density distribution f (x, p) of beam particles in a 6-dimensional phase space comprised of the variables (x, px , y, py , z, pz ), in which (xi , yi , zi ) = xi are the spatial coordinates of the particle locations and (px,i, py,i , pz,i) = pi are the momenta of the particles

3

along the corresponding directions. It is assumed that the distribution function is initially a separable (orthogonal dimensions are uncorrelated) equilibrium solution to the Vlasov equation [5]. Two-dimensional projections of the phase space are often examined separately, with matching directions paired off in trace spaces such as the one diagrammed in Fig. 1.1a. This plot illustrates a typical ideal bi-Gaussian beam distribution in the xx′ trace space of the form  ′2   2 −x −x 1 ′ , exp exp fx (x, x ) = 2 2πσx σx′ 2σx 2σx2′

(1.1)

where the quantities σx2 and σx2′ are the second moments of the transverse beam distribution given by σx2

=

Z

∞

Z

∞

−∞

σx2′

=

−∞

Z Z

∞

x2 fx (x, x′ )dxdx′

(1.2)

x′2 fx (x, x′ )dxdx′ .

(1.3)

−∞ ∞ −∞

The elliptical contours in the plots in Fig. 1.1 represent constant values of fx . It is customary to define an RMS trace space ellipse, written in terms of the second moments as x2 x′2 + = 1, σx2 σx2′

(1.4)

as an envelope encompassing a representative fraction of the beam particles; the evolution of this ellipse is a useful representation of the dynamics of a beam as it travels along an accelerator. It is convenient to parameterize the RMS ellipse in terms of the Twiss parameters (εx , βx , αx , γx ) as γx (z)x2 + 2αx (z)xx′ + βx (z)x′2 = εx ,

(1.5)

where z is the longitudinal position along the beamline. The Twiss parameters are related to the RMS envelope as illustrated in Fig. 1.1b. The parameter εx

4

RMS ellipse

α/β

√ γε √ε/β

x’ 0

x’ 0 0

x

x’

√ε/γ

√ βε

x 0

x (a) Figure 1.1

(b) correlated state

uncorrelated state

Examples of bi-Gaussian beam distribution and 2D trace space pro-

jections for (a) uncorrelated state and (b) correlated state.

is called the emittance, and represents the area in trace space A ≡ πεx enclosed by the RMS ellispe. Although the actual area of the beam distribution in phase space cannot change (Liouville Theorem [7]), the area of the RMS ellipse, which defines an effective area of the beam, can change. If the phase space distribution of a beam is distorted by nonlinear forces, the ellipse that encloses an RMS fraction of the distribution must be redrawn and has a larger area. This represents irreversible emittance growth, and is a catastrophic event in terms of the beam quality. Emittance is an extremely important figure of merit describing the quality of a beam; it quantifies the tendency of a beam to diverge away from a focus, in much the same way as the wavelength of a Gaussian photon beam determines its diffractive behavior. The emittance of a beam has a fundamental lower limit imposed by the finite temperature of the electron plasma, and care must be taken to control its growth as the beam experiences accelerating and focusing forces. The production of low-emittance beams has been a technological challenge and the subject of ongoing, intensive research efforts [8, 9, 10, 11, 12]. The parameter αx describes the correlation between the trace space variables

5

x and x′ . An αx value of zero occurs when the beam is at a focal point and in an uncorrelated state; this condition is referred to as a beam waist. As a beam passes through force-free drifts and focusing elements, it transitions to and from uncorrelated waists; Fig. 1.1 illustrates trace space ellipses for both correlated and uncorrelated states. In another analogy to Gaussian optics, the parameter βx (also called the βfunction) plays the role of the Rayleigh length, representing the distance over √ which the beam size σx grows by a factor of 2. The β-function at a waist, denoted β ∗ , is given by β∗ =

σx2 |z=z0 εx |z=z0

(1.6)

where z0 is the longitudinal location of the waist. This quantity is useful for evaluating the ability of a focused beam to pass through a limiting aperture of a given length. As a beam’s electric current and its density in 6D phase space increase, it is said that the brightness of the beam increases. High-brightness beams, of the variety produced for use in the experiment discussed in this dissertation, are space-charge dominated, meaning that their dynamics are influenced chiefly by the internal forces between the beam electrons. In contrast, the motion of an emittance-dominated beam is dictated by its emittance. The effects of emittance, space charge, and focusing on beam dynamics are included as the inhomogenous terms in the RMS envelope equation, given by σx′′

ε2x,rms hxx′′ i γ′ ′ + σx = + . γ σx3 σx

(1.7)

The first and second terms on the right-hand side represent the effects of the emittance and the transverse forces respectively. The transverse force term can describe commonly included effects such as space charge, magnetic optic focusing, and alternating-gradient RF field focusing [13]. This equation is derived by

6

writing the derivatives of the RMS beam size in terms of the beam distribution moments, and describes the evolution of the beam envelope in a straightforward mathematical form that makes it simple to visualize the separate effects of space charge and emittance. A matrix notation is often more useful for actually calculating the evolution of the beam along an accelerator. For each of the three 2D trace spaces comprising the full 6D phase space, there is a matrix defined in terms of either the second moments or the Twiss parameters as     σxx σxx′ βx −αx  = εx  . σ= σx′ x σx′ x′ −αx γx

(1.8)

The σ matrices for the three orthogonal planes can be collected into a single 6×6 block-diagonal matrix Σ. In general, Σ loses its diagonality under the influence of bend magnets or a solenoid, which cause the transverse and longitudinal phase planes to intertwine. A transport matrix R describes a linear transformation associated with an element of the accelerator such as a drift, focusing magnet, or bend magnet, and is applied at each step along the beam trajectory.

1.1.2

Radiation from Electron Beams

Following an initial overview of radiating electric charges, the discussion in this section is limited to types of beam-based radiation that will be discussed later on in this study, namely Cherenkov radiation and transition radiation. From a fundamental standpoint, an electron (or any charged particle) will radiate when it is accelerated. This well-known fact can be understood intuitively with a simple physical picture. Consider an electron traveling with a constant velocity. Regardless of its speed, an inertial reference frame can be found in which the electron and its radially extending electric field are at rest. If the electron is

7

particle (retarded position)

β

. β z

n

r observation point

x Figure 1.2

x

y

Coordinate system used in calculating radiation by a moving charge.

instead accelerating, no inertial frame exists that will preserve the static form of the surrounding fields. In this situation the field line pattern is disturbed as the fields reorganize, and this disturbance propagates outward at speed c as radiation. In a more colloquial illustration, one might imagine that as a particle experiences a sudden force its field lines are shaken by a “whiplash” effect that travels out to infinity. Detailed and illuminating illustrations of electric field lines of accelerating particles can be found in [14]. For the case of a single electron in free space experiencing acceleration, the differential radiated intensity is given by [15] Z 2 e2 ω 2 ∞ d2 I iω(t−n·r(t)/c) = 2 n × (n × β)e dt , dωdΩ 4π c −∞

(1.9)

referring to the coordinate system defined in Fig. 1.2. This spectral angular

distribution is an important quantity in the formulation of radiation problems, giving the total radiated intensity when integrated over all angles and frequencies. A particularly distinct feature of the angular radiation pattern from an accelerating charge in relativistic motion is its intensity peak at an angle of 1/2γ radians from the direction of instantaneous velocity, where γ is the standard Lorentz relativistic factor. This general formulation is used to study radiation produced by charges under many forms of acceleration. The rest of this section will be spent

8

Incoherent

>λ ~λ

λ

=

Coherent

Figure 1.3

Illustration of the coherent radiation process in an electron bunch.

briefly discussing specific cases that are of wide interest in accelerator physics and have particular relevance to this dissertation.

1.1.2.1

Coherently Radiating Beams

In general, the total radiation intensity emitted by a group of electrons is the sum of the intensities from each individual particle. When the electrons are grouped together or otherwise arranged in a distribution with some harmonic content, the radiation field they emit can add constructively, as illustrated in Fig. 1.3. The term coherent is used throughout this dissertation to describe radiation processes in which the emitting particles are radiating in cooperation with each other. If P (ω) represents the power radiated by a single electron, the power radiated by a distribution of N electrons is given by [16, 17] PN (ω) = [N + N(N − 1) F (ω)] P (ω).

(1.10)

We see that the limit F (ω) → 0 represents completely incoherent excitation, where the radiated power is just the sum of that from each of the N electrons. In the limit F (ω) → 1, the power is N 2 times P (ω), enhanced by a factor of N

9

over the incoherent case. In an intense beam, N is a very large factor and can be on the order of 109 − 1010 . The degree of coherence is quantified by the form factor F (ω), which is the squared Fourier transform of the charge distribution of the radiating electron beam: Z F (ω) =

∞

iω t

ρ(t)e

−∞

For an ideal Gaussian charge distribution ρ(t) =

1 √

σt 2π

2 dt .

2 /2σ 2 t

e−t

,

(1.11)

(1.12)

the form factor is also Gaussian: 2

2

F (ω) = e−σt ω .

(1.13)

We introduce a characteristic roll-off wavelength λr , quantifying the wavelength below which the coherence of the excitation is greatly diminished, defined as the λ value at which the form factor is reduced by a factor of e: 2

2

F (λr ) ∝ e−1 = e−σz (2π/λr ) .

(1.14)

This condition gives a relationship between the electron bunch length and λr , λr = 2 πσz ,

(1.15)

which represents a lower bound on the wavelengths of radiation that may be coherently excited by a beam of length σz .

1.1.2.2

Cherenkov Radiation

A distinct and particularly useful form of radiation caused by charged particles is that of Cherenkov radiation, named for Paul Cherenkov who did the first

10

-

-

+ +

+

e-

-

-+ -+ + Figure 1.4

Illustration of atoms polarized by a passing charge.

extensive experimental investigation of the phenomenon in the 1930s [1, 18, 19]. Radiation of this type is emitted when a charged particle travels in or near a dielectric medium at a speed greater than the speed of light in the material. A distinguishing feature is that the radiation is emitted at an angle to the particle trajectory axis that is determined by the speed of the particle and the refractive index of the medium. The phenomenon is at the heart of this dissertation study, and is used widely throughout modern physics in experimental apparatus [20, 21, 22]. Cherenkov radiation can be understood in a basic way with a simple physical picture [23]. Consider the illustration in Fig. 1.4. As a charged particle such as an electron travels in a dielectric medium, its electric field acts on the nearby constituent atoms, polarizing them by pulling the positively charged nucleus toward the driving particle while repelling the negative valence electrons. The degree of this effect depends on the polarizability of the medium, quantified by the permittivity ε. At low velocities this effect is completely symmetric in all directions, following the form of the electron’s field, and therefore any polarization field cancels at long range and there is no radiation. If instead the electron moves at a speed comparable to the speed of light in the medium, the relaxation of the polarized atoms begins to lag behind the passage of the driving particle.

11

This introduces an asymmetry in the axial direction along the electron trajectory, which results in a net dipole field that does not cancel even at large distances. This gives rise to radiation pulses originating at each point along the trajectory as the electron passes. However, the wavelets from all points along the trajectory combine destructively so that there is no net radiation field at a long distance. If the electron velocity exceeds the phase velocity c/n of light in the dielectric, wavelets radiated from every point along the trajectory can add constructively, forming a reinforced phase front that propagates away from the trajectory axis at an angle set by the dielectric constant of the medium and the particle velocity. This Huygens’ wavelet model [24], analogous to the “sonic boom” phenomenon seen in sound waves, is illustrated in Fig. 1.5. The Cherenkov angle θC is given by cosθC =

1 , βn

(1.16)

where n is the refractive index and β = v/c is the normalized electron velocity. The Cherenkov condition, defining the electron velocity threshold below which there is no radiation, is then β = 1/n. The maximum emission angle, reached at the relativistic velocity limit β = 1, is given by θmax = cos−1 (1/n). The analytical theory of Cherenkov radiation was first outlined by Frank and Tamm [25, 26], somewhat concurrently with Cherenkov’s early experiments. They simplify the problem by assuming an unbounded system with an infinite particle trajectory, a continuous medium characterized only by the dielectric constant, and a constant particle velocity. They find that the radiated energy, per unit frequency interval and particle path length, increases linearly with the frequency of radiation in a continuous spectrum corresponding to the “delta function” nature of the single-particle excitation. As a real medium is always dispersive, i.e. n is a function of frequency, the spectrum is limited to those frequency values for

12

cT n

e

θC βcT

dielectric

Figure 1.5

Illustration of Huygens’ wavelet model of Cherenkov radiation.

which the Cherenkov condition is satisfied. Of particular relevance to this study is the case of a charged particle passing close to, but not inside, a dielectric medium. Atoms in the material are still subject to the influence of the electric field of the passing particle, and Cherenkov radiation is produced in the usual manner. However, the yield at a given wavelength falls off quickly as the distance between the particle and the medium becomes a significant fraction of that wavelength. A critical advantage of this scenario is that it eliminates particle energy loss due to ionization in the medium, which would limit the particle energy available for radiation production to a small fraction of the total. The case of Cherenkov radiation from a particle moving down the vacuum channel in a cylindrical dielectric-lined waveguide was first treated theoretically by Abele in 1952 [27]. In contrast to the continuous spectrum of Cherenkov radiation in an unbounded system, he finds that the radiated energy is concentrated in discrete modes corresponding to the eigenmodes of the structure. It is seen that the radiation yield increases with frequency at a rate

13

slower than the linear frequency dependence of the case of no vacuum channel.

1.1.2.3

Transition and Diffraction Radiation

When an electron passes through a discontinuous interface between two media, radiation is emitted as its fields reorganize. This phenomenon is termed transition radiation (TR) [28, 15]. In an electron accelerator, TR is typically seen when a relativistic electron beam strikes a piece of metal such as a foil or mirror [29]. In this context, the event can be viewed as a collision between the beam and its image charges at the conducting surface. The forward angular intensity distribution is azimuthally symmetric about the electron velocity direction, with a hollow “donut” transverse shape peaked at an angle 1/γ from the axis. The radiation is radially polarized. In the case of a particle striking a conducting surface, an identical “backward” pattern is also produced in the direction of specular reflection from the surface [30]. Due to the “delta-function” nature of the interaction, the TR spectrum for a single particle is flat over a very broad range of frequencies. As such, incoherent TR at optical wavelengths (OTR) from a beam of electrons is often used to view a beam spot on a metal foil screen. The coherent transition radiation (CTR) spectrum contains information about the transverse and longitudinal shape of the beam, and has become a standard beam diagnostic tool [31, 32, 33, 34, 35, 36]. It is found that the scenario of a beam passing through an aperture in a metal foil produces a similar radiation pattern due to the diffraction of the beam’s electric field from the aperture [37, 38]. Diffraction radiation (DR) indeed reduces to the familiar TR in the limit that the size of the aperture shrinks to zero [39, 40, 41]. Interest in coherent diffraction radiation (CDR) as a beam diagnostic alternative to CTR has grown due to its potentially non-destructive nature [42,

14

43, 44]. In the study at hand, we are primarily interested in CDR as a source of background radiation due to the small apertures in the beam path in the vicinity of the experimental interaction point.

1.2

Wakefield Physics

As a relativistic beam of electrons travels along an accelerator, its Lorentzcontracted electric field terminates on the walls of the vacuum pipe or on nearby structures. Geometric discontinuities and/or finite conductivity in the surroundings cause these fields to be “scraped off” and travel away as electromagnetic radiation [45]. These radiation fields are termed wakefields because they are reminiscent of the wakes left behind a speedboat in water. In keeping with this analogy, wakefield radiation can exist only behind the relativistic driving beam due to causality restrictions. A wakefield analysis formalism is relevant to this dissertation because the Cherenkov radiation produced by a beam moving through a DLW structure can alternatively be understood as a wakefield. This approach allows a powerful way of understanding and calculating the energy lost by the driving bunch as well as the radiated power flow along the structure. It is convenient to approach the conceptual understanding of wakefields from a frequency-domain standpoint. In an analogy to Ohm’s Law, we can write V (ω) = −Z(ω)I(ω),

(1.17)

where I(ω) is the Fourier transform of the electron beam current profile and V (ω) represents the voltage it induces as it interacts with its surroundings. The proportionality Z(ω) between these two quantities is the wake impedance, and its form is dictated solely by the geometric and material properties of the surrounding environment [7]. For example, if the beam were passing through a pipe

15

r

z=0

Figure 1.6

+s

s=0

trailing particle

driving particle

z

Coordinate system used in wakefield analysis.

with a sinusoidally-varying wall radius, the impedance would be a narrow-band function of frequency. A beam that is microbunched at a similar frequency would experience very strong coupling to wakefields in this scenario. If there is a step discontinuity in the pipe wall, such as when the beam passes into a larger-sized pipe, the impedance has a broadband structure. We are primarily interested in the integrated effect of a wakefield on a test particle as it follows a driving particle along a structure. We confine our discussion to the case in which the driving and test particles travel identical trajectories lying on the structure axis, using the coordinate system defined in Fig. 1.6, and consider only longitudinal forces. This scenario has been used to experimentally map the form of longitudinal wakefields [46, 47]. We define a longitudinal wake potential Wz to represent the change in electric potential, per unit driving charge, of a test particle that has followed the driving particle at a constant distance s for the entire length L of a structure, given by [48]   Z z+s 1 L dz, Ez z, t = Wz (s) = − q 0 c

(1.18)

where Ez is the longitudinal electric field of the wake on the axis of the structure. By convolving this single-particle response with an arbitrary axial driving beam

16

charge distribution ρ(s) we obtain Z s V (s) = Wz (s − s′ ) ρ(s′ ) ds′ ,

(1.19)

−∞

which is the potential difference experienced along a structure by a test particle at a point s in or behind an arbitrary driving electron bunch. Because it is valid for particles within the driving beam itself, Eq. 1.19 can be used to calculate the total energy U lost to self-wakefields by a driving bunch as it traverses a structure. This is accomplished by integrating V over the entire charge distribution, Z ∞ U= V (s) ρ(s) ds. (1.20) −∞

A particularly useful concept is the wakefield theorem 1 , which states that the maximum longitudinal electric field in a wake behind a driving particle is twice the decelerating field Ez,dec experienced by the particle [48, 49]. The same is true in general for an extended beam charge distribution, although it is possible in specific scenarios to produce an accelerating field larger than 2Ez,dec [50, 51, 52]. The theorem can be proven using energy conservation arguments, and can be a useful deduction tool in wakefield calculations. In the case of wakefields in a waveguide, the power flow P along the structure is calculated by integrating the longitudinal component of the time-averaged Poynting vector S = 21 E × H∗ over the cross section of the structure: Z Z P = S · ˆz dA.

(1.21)

A

The energy stored in the EM field is given by Z Z Z
1 uEM = ε|E|2 + µ|H|2 dV. V 2

(1.22)

We will employ this formalism in the next chapter to calculate the radiated energy, group velocity, and pulse length of coherent Cherenkov radiation wakefields in the dielectric-lined waveguide structure under study. 1

Also called the fundamental theorem of beam loading.

17

1.3

Terahertz Radiation

The THz region of the EM spectrum lies between far infrared and microwave radiation, and is typically defined as the frequency range 100 GHz − 10 THz. Intensive research exploring methods of generating THz radiation continues to progress successfully [53, 54], and the “THz gap” is becoming increasingly shallow. Methods based in optics and photonics are receiving wide attention and use, such as the quantum cascade laser (QCL) [55, 56], the optically-pumped THz laser (OPTL), the Auston switch [57], and frequency mixing [58]. A particularly interesting study in 1966 reported operation of a water-vapor gas laser at 118 µm wavelength [59]. Electron-beam-based sources have proven to be effective at producing very high peak and average power and providing wide tunability; these sources include free-electron lasers, synchrotron light sources [60, 61], Cherenkov masers [62], and Smith-Purcell radiators [63, 64]. Among the foremost applications of THz are imaging and spectroscopy. The utility of THz light for imaging is due to its ability to penetrate common materials such as paper, cloth, and wood [65, 66, 67], and to do so with reduced susceptibility to Rayleigh scattering relative to infrared light [68]. Thus it is envisioned for use in security screening, as a metal weapon could be seen under a person’s clothing or contraband detected inside wooden shipping crates. It is particularly attractive in this regard because it is non-ionizing, and therefore may be safe to use on humans in situations where X-rays would prove harmful. THz radiation has been employed in spectroscopy for over a century under the term far-infrared spectroscopy [69]. Many organic molecules have vibrational modes at THz frequencies [70], including amino acids [71], proteins [72], and DNA [73]. In addition to medical applications such as tissue characterization, THz

18

spectroscopy has potential for identifying biochemical security threats as many chemical molecules also possess distinct THz absorption spectra. Other applications include atmospheric characterization and manufacturing quality control.

19

CHAPTER

2

Wakefields in a Cylindrical Dielectric-Lined Waveguide

The electromagnetic structure studied in this dissertation is a hollow circular cylindrical dielectric tube coated on the outer surface with metal to form a dielectriclined waveguide (DLW). As a relativistic electron bunch travels along the vacuum channel in the structure, coherent Cherenkov radiation (CCR) produced in the dielectric material propagates in a discrete set of modes due to the presence of the conducting outer boundary of the waveguide. The beam exchanges energy with the modes via its deceleration by the longitudinal electric field component. The DLW behaves as a slow-wave structure, supporting modes with phase velocity equal to the beam velocity. The modes are thus capable of efficient energy exchange with the beam, because the coupling field can remain in phase with the beam along the entire length of the structure. With a sufficiently short (RMS bunch length σz less than a radiation wavelength λ) driving beam containing N electrons, the radiation process is coherent and the radiation energy at the relevant longer wavelengths is enhanced by a factor proportional to N over incoherent radiation. This coherent excitation process can provide quite high radiation power, offering a simple and effective energy conversion scheme and allowing creation of sources producing unprecedented peak power in the THz

20

spectral region. This chapter presents the physical concepts relevant to the operation of a DLW-based radiation source in a mathematical formalism useful for making predictions that will be compared with experimental results. The analysis is compared to computer simulation results and seen to agree very well. As the goal of this study is comparison with experimental results, we refrain from discussing here certain physical aspects of the scenario that, while interesting, were either not measured or not of particular consequence in the experiment. Among these are polarization, reflection, parasitic mode conversion, and resistive wall losses, to name a few.

2.0.1

Experimental Background

Dielectric wakefield (DW) devices have been studied experimentally in several similar capacities for the past few decades. Presented here is an overview of advancements that highlight the consistent interest in and potential of these simple structures. An early application of a DLW, in a transverse deflecting mode, was as a traveling-wave particle separator, used to select particles of a certain mass out of a beam [74] to be directed to a fixed-target high-energy physics experiment. This is an example of a synchronous device, in which external RF power is fed into a structure designed for use with particles of a specific velocity; in the case of the separator, unwanted particles travel at velocities different than the design RF phase velocity and thus experience phase slippage. Only the particles near the design velocity would then be efficiently steered in the correct direction. Another example of a synchronous device is the Cherenkov maser, also called Cherenkov free-electron laser (FEL) or Cherenkov free-electron maser (FEM). In this case, the input RF power acts as a seed to microbunch a continuous (or

21

long-bunch) electron beam, which in turn coherently radiates wakefields in a gain process that amplifies the original frequency. Cherenkov FELs are also operated in a self-amplified configuration, in which there is no external power source and the beam microbunching instability is driven solely by spontaneous emission of wakefields arising from spectral noise in the beam charge distribution. These devices have demonstrated efficient power amplification at microwave and THz frequencies [62, 75]. Self-amplified devices have the advantage that they eliminate the need for a high-power external seed source, which is a particularly limiting factor at higher frequencies approaching THz. The need for high-gradient accelerator technology has prompted research into the use of wakefields for the acceleration of electrons. Proven scenarios include plasma wakefield acceleration (PWFA) [76, 47, 77, 78, 79, 80, 81] and laser wakefield acceleration (LWA) [82, 83, 84], and not absent from the list is the dielectric wakefield accelerator (DWA) [46]. In alternative to using wakefields directly to accelerate particles, they may be harnessed as a power source to drive external accelerator modules [85]. The DWA structure is in some cases identical to the DLWs used for radiation production, except its design has an eye toward creating wakefields behind the driving beam tailored for imparting energy to a trailing electron bunch. The implementation of this technique is affected greatly by the fundamental limit on the transformer ratio achievable with a conventional symmetric driving beam, as described in Section 1.2. Experiments exploring the use of bunch trains and ramped current profiles for driving wakefields have demonstrated significant transformer ratio enhancement, addressing a central hurdle in the practical realization of machines based on wakefield acceleration [86, 87]. A limiting factor on achievable gradients in conventional accelerator technology is electric breakdown; as state-of-the-art high-energy beams can produce wakefields on the order of several GV/m in a dielectric structure, breakdown again becomes

22

a concern. A recent study quantified the breakdown field threshold and beam current tolerable in a specific fused silica DWA structure, providing valuable insight into the limitations that can be expected as the technology matures [88]. Among the experimental milestones enumerated above, the self-amplified Cherenkov FEL is the closest cousin to the scenario under investigation in this dissertation and as such a closer examination of its operating principles is warranted. The process relies on an instability that begins as random noise in the microscopic beam charge distribution and grows as a result of the feedback of incoherent wakefields onto the beam. As incoherent Cherenkov wakefields driven by a relatively long (compared to a radiation wavelength) electron bunch act back on the driving beam, the electrons begin to organize into microbunches, imparting a modulation on the beam that is periodic at the radiation wavelength. With this small modulation present, the wakefield radiation process begins to become coherent at one or more DLW mode wavelengths, which in turn modulates the beam more strongly and leads to increasing coherence. Ultimately, the beam is strongly microbunched and radiates coherently at a significantly enhanced level compared to the incoherent case. This gain process is analogous to the selfamplified spontaneous emission (SASE) operation regime employed in high-gain FELs [89]. The experiment described in this dissertation is distinct from the work outlined in the previous paragraphs in that it utilizes a short-driving-bunch wakefield scenario, such as is common for DWA operation, for the purpose of narrow-band radiation production. The scenario is based solely on the coherent excitation of wakefields, not on a gain process or microbunching instability, a regime often referred to as superradiance in the context of pre-bunched beam FEL operation [90, 91, 92]. The place of this work in the broader context of previous achieve-

23

ments is chiefly as an adaptation of the Cherenkov maser/FEL concept to the realm of ultra-short electron beams.

2.1

Analytical Model

Analytical models of beam-driven EM radiation in DLW structures are welldeveloped, and several different approaches are presented in the literature. One typical approach uses the superposition of a Green’s function solution for fields in an unbounded space with a general solution of the homogenous wave equation subject to the structure boundary conditions [93, 94]. Another method represents the field solution as an infinite sum over eigenmodes and exploits orthogonality conditions to solve for the mode coefficients [95]. For the case of multiple layers of different dielectric materials, a matrix method has been developed to allow an algorithmic solution for an arbitrary number of layers [96, 97, 98, 99] . The purpose of our analysis is to gain insight into the dispersion characteristics and transverse modal structure of the wakefields, as well as to calculate the mode frequencies and radiated energy for comparison with measurements. In the case that the wavelengths of interest are smaller than the dimensions of the structure, a ray optics model is useful for deriving a simple resonance condition describing the guided modes. Such a highly overmoded analysis is commonly applied to understand, e.g., the properties of fiber optics and dielectric optical waveguides [100]. This type of model has been successfully employed in the context of a beam-driven DLW in the work of Thompson et al. [88, 101]; this work also uses a heuristic Gauss’ Law model to estimate the electric fields driven by a high-energy electron beam with reasonable accuracy, as corroborated by simulation. This convenient method was found to be inadequate for use in the present study because the wavelengths of interest, belonging only to the

24

fundamental mode, are on the order of the size of the structure and ray optics does not apply. Therefore, we must rely on a more formal analytical treatment yielding unwieldy formulas that require numerical solutions. Nevertheless, the results are accurate and give satisfying agreement with measurements and simulation, as will be discussed in Chapter 4. The primary simplifying assumption used in this analysis is that the device operation is single-mode; that is, while the structure can in principle support many modes, only the lowest frequency modes (shown to be the TM01 and HEM11 modes) have wavelengths long enough to be excited by our beam according to the coherence roll-off condition presented in Section 1.1.2.1. Of these, the TM01 mode couples most strongly to the beam through its relatively large on-axis longitudinal electric field and thus dominates the excitation. This assumption is justified by simulation results and confirmed by our measurements. Simplification in this manner allows us to take a more targeted approach in our analysis, finding the relevant mode frequencies and calculating quantities of interest such as radiated energy and group velocity on a single-mode basis. The first step in our analysis is to calculate the eigenmodes of the structure in the absence of a driving source. Energy balance arguments will then be used to deduce the wakefield response of the structure to a point charge. Finally, this point source response will be convolved with an idealized beam charge density function to give the full beam-driven wakefield, which is used to calculate the total energy radiated as the beam travels the length of the structure.

2.1.1

Eigenmodes in a Dielectric-Lined Waveguide

We make use of standard cylindrical coordinates (r, z, θ). The coordinate system and the parameters of the DLW tube structure are illustrated in Fig. 2.1. We

25

b a

Conductor

θ

eDielectric ,

r z

ε

L Figure 2.1

Coordinate system used in analytical calculation of wakefields.

will approach the solution to this problem in terms of fields instead of potentials. This is the simplest method for a general cylindrical waveguide problem, because the transverse fields in a waveguide can be expressed in terms of the longitudinal fields [15] and thus there are only two field component equations (for Ez and Bz ) to be solved. Assuming that all fields in the waveguide have z and time dependences of ei(kz z−ωt) , the Maxwell equations produce a two-dimensional wave equation    2  E  ∇t + ν 2 =0 (2.1) B where ν 2 = µǫω 2 − kz2 represents the transverse wavenumber. Note that this differential equation describes the variation of the 3D fields (E, B) on the 2D cross section of the waveguide, and can be separated into three identical equations for the scalar field components. This equation is derived from Maxwell’s equations in detail in Appendix B. Writing the transverse Laplacian operator ∇2t in cylindrical coordinates, the equation governing the longitudinal field components becomes      2 2 Ez 1 ∂ 1 ∂ ∂ 2 = 0. (2.2) + + + ν B  ∂r 2 r ∂r r 2 ∂θ2 z

26

We now proceed generically for either Ez or Bz , defining     Ez  RE (r) ≡ Θ(θ) = R(r)Θ(θ) B  R (r) z

(2.3)

B

and using a standard separation of variables to reduce the partial differential equation (PDE) to two ordinary differential equations (ODE): d2 Θ + m2 Θ = 0 dθ2   d2 R 1 dR m2 2 + + ν − 2 R = 0. dr 2 r dr r

(2.4) (2.5)

Note that each ODE is second order and therefore has two linearly independent classes of solutions. The solutions to Eq. 2.4 are clearly sine and cosine functions. We can recast Eq. 2.5 into a familiar form by the change of variable νr → x, yielding the Bessel equation   d2 R(x) 1 dR(x) m2 + + 1 − 2 R(x) = 0. dx2 x dx x

(2.6)

The solution to this equation is written in terms of the Bessel functions of the first and second kinds, Jm (x) and Ym (x), respectively [102]. For the special case of a purely imaginary x, the solutions are the modified Bessel functions of the first and second kinds, Im (x) and Km (x). Since we have as of yet made no restriction on the transverse wavenumber ν, it may take on positive, negative, or imaginary values. m represents the order of the Bessel function as well as the angular frequency of the azimuthal function Θ(θ).

2.1.1.1

Field Solutions

The general solution to Eq. 2.4 is Θ(θ) = A1 sin(mθ) + A2 cos(mθ).

27

(2.7)

The sine function solutions have the undesirable feature of being zero for m = 0. Since it would be preferable to have m = 0 denote no azimuthal dependence (Θ(θ) = 1), we will keep only the cosine functions, and for convenience use complex exponential notation Θ(θ) = eimθ ,

(2.8)

where it is understood that we are concerned only with the real part of the solution. We have dropped the constant A2 , choosing instead to absorb it later into the overall field amplitude. It is clear that m may take on only integer values; otherwise the azimuthal variation of the fields is not single-valued, which is an unphysical result. We will henceforth rely on Eq. 2.8 to represent the azimuthal variation of all field components. The general solutions to Eq. 2.5 are  :0  C1 Im (kr) + C2Km (kr) for 0 < r < a  R(r) =  C J (κr) + C Y (κr) for a < r < b . 3 m

(2.9)

4 m

The constant C2 has been set to zero because the function Km (kr) is infinite at r = 0 in the vacuum region. We have adopted the notation   −k 2 = µ0 ε0 ω 2 − k 2 for 0 < r < a z ν2 ≡ κ2 = µ ε ε ω 2 − k 2 for a < r < b 0 0 r z

(2.10)

to represent the forms of the radial wavenumber in the two different media, where the relative permittivity εr is the ratio of the permittivity in the dielectric to the free-space value ε0 . Throughout this analysis we assume that the dielectric material is non-magnetic, i.e. its permeability is the same as the free-space value µ0 . We have chosen a purely imaginary value ν = ik in the vacuum channel, due to the sub-luminal phase velocity in this region.

28

The next step is to determine the constants C1 , C3 , and C4 . This must be done for each field (Ez , Bz ) in turn by utilizing appropriate boundary conditions. We will first work with Ez and then repeat the process for Bz . We can make an initial simplification by assigning the constant C1 to represent an overall amplitude of the field, replacing it with the symbol E0 . It is well known [15] that the component of electric field parallel to an interface between two media is continuous across the interface. It follows that the component parallel to a perfectly conducting surface must be zero at the surface, since all fields must be zero inside the conductor. The boundary conditions on Ez are then Ez is continuous at r = a

(2.11)

Ez = 0 at r = b.

(2.12)

Applying these to Eqs. 2.9 yields the system of equations E0 Im (ka) = C3,E Jm (κa) + C4,E Ym (κa)

(2.13)

0 = C3,E Jm (κb) + C4,E Ym (κb),

(2.14)

which is simple to solve for the two unknown constants: C3,E =

−E0 Im (ka)Ym (κb) Ym (κa)Jm (κb) − Ym (κb)Jm (κa)

(2.15)

C4,E =

E0 Im (ka)Jm (κb) . Ym (κa)Jm (κb) − Ym (κb)Jm (κa)

(2.16)

In order to simplify the notation we define αm ≡

Im (ka) , Ym (κa)Jm (κb) − Ym (κb)Jm (κa)

(2.17)

and we can write the solution for the longitudinal electric field radial function as

RE (r) =

 

E0 Im (kr)

for 0 < r < a

E α [J (κb)Y (κr) − Y (κb)J (κr)] for a < r < b . 0 m m m m m 29

(2.18)

The complete solution for the field Ez is then Ez (z, r, θ, t) = RE (r)eimθ ei(kz z−ωt) .

(2.19)

We will now return to Eq. 2.9 and match a different set of boundary conditions to find the radial function RB (r) describing the longitudinal magnetic field Bz . The appropriate conditions on the magnetic field in this case are Bz is continuous at r = a

(2.20)

∂Bz = 0 at r = b. ∂r

(2.21)

Applying these conditions yields the equations B0 Im (ka) = C3,B Jm (κa) + C4,B Ym (κa) ′ 0 = C3,B κJm (κb) + C4,B κYm′ (κb),

(2.22) (2.23)

where we have introduced the overall field amplitude B0 as described above in the case of Ez . Note that a prime (′ ) henceforth denotes differentiation with respect to the whole argument of the Bessel function, since the constant κ has already been considered when performing the derivative with respect to r. Again, the solution of this equation for the two unknowns is straightforward: C3,B =

−B0 Im (ka)Ym′ (κb) ′ (κb) − Y ′ (κb)J (κa) Ym (κa)Jm m m

(2.24)

C4,B =

′ B0 Im (ka)Jm (κb) . ′ ′ Ym (κa)Jm (κb) − Ym (κb)Jm (κa)

(2.25)

Defining the constant ηm ≡

Im (ka) , ′ Ym (κa)Jm (κb) − Ym′ (κb)Jm (κa)

30

(2.26)

we write the solution for RB (r) as   B0 Im (kr) for 0 < r < a RB (r) = B η [J ′ (κb)Y (κr) − Y ′ (κb)J (κr)] for a < r < b 0 m m m m m

(2.27)

and the complete form of the longitudinal magnetic field is Bz (z, r, θ, t) = RB (r)eimθ ei(kz z−ωt) .

(2.28)

The longitudinal field solutions given in Eqs. 2.19 and 2.28 can now be used to calculate the transverse vector components of the fields. The transverse fields are given in terms of the longitudinal components by [15] Et = Bt =

i µεω 2

− kz2

i µεω 2

− kz2

[kz ∇t Ez − ωˆ z × ∇t Bz ] [kz ∇t Bz + µεωˆ z × ∇t Ez ] .

(2.29)

Writing the two-dimensional gradient operator ∇t in cylindrical coordinates, carrying out the cross product, and inserting the explicit forms of Ez and Bz , we obtain the transverse field components:   i dRE (r) ω Er = kz + imRB (r) eimθ ei(kz z−ωt) µεω 2 − kz2 dr r   kz dRB (r) imθ i(kz z−ωt) i e e imRE (r) − ω Eθ = µεω 2 − kz2 r dr   i dRB (r) µεω Br = kz + imRE (r) eimθ ei(kz z−ωt) µεω 2 − kz2 dr r   kz dRE (r) imθ i(kz z−ωt) i e e . imRB (r) + µεω Bθ = µεω 2 − kz2 r dr

(2.30) (2.31) (2.32) (2.33)

We can relate the overall field amplitudes E0 and B0 to each other for specific cases of interest by noting that, following Jackson, Eqs. 2.29 imply Bt =

µ ˆ z × Et Z

31

(2.34)

where Z is the wave impedance given by   kz when Bz = 0 Z = εω  µω when E = 0. z kz

(2.35)

The special cases Bz = 0 and Ez = 0 correspond to the transverse magnetic

(TM) and transverse electric (TE) modes respectively, as discussed in a later section. The application of the wave impedance will allow the fields in the DLW to be determined to within a single arbitrary constant E0 , which is ultimately determined by the driving source of the field excitation.

2.1.1.2

Dispersion Relation

While Eqs. 2.19, 2.28, and 2.30 – 2.33 represent the full functional form of the fields in the DLW up to an unspecified overall amplitude, we have not yet determined the discrete spectrum of wavenumbers that are able to propagate in the structure. By imposing boundary conditions on the transverse field components, we can obtain a dispersion relation that will describe the spatial characteristics and frequencies of radiation modes present in the waveguide. Although boundary conditions apply to each of the four transverse field components at both interfaces (r = a and r = b), we need only consider two of these in order to uniquely constrain the problem. The remaining conditions will then be automatically satisfied, by virtue of their coupling to the already-confined longitudinal field components. We choose to make use of the conditions on the components Er and Eθ at the inner edge of the dielectric. The radial component of the electric displacement field D = ε0 εr E must be continuous across the interface between two media, leading to the condition Er |ra

32

(2.36)

which we apply to Eq. 2.30. After calculating the radial derivative of RE (r) and being careful to insert the appropriate forms of the radial wavenumber for each region, we obtain the equation  ′  −Im (ka) εr ′ ′ E0 kz − αm {Jm (κb)Ym (κa) − Ym (κb)Jm (κa)} k κ   ω −1 εr + B0 im − 2 Im (ka) = 0. a k2 κ

(2.37)

Similarly, by requiring that Eθ be continuous across r = a we arrive at a second equation ′ 1 Im (ka) ′ ′ ′ B0 ω ηm {Jm (κb)Ym (κa) − Ym (κb)Jm (κa)} + κ k   1 kz −1 − 2 Im (ka) = 0. + E0 im a k2 κ



 (2.38)

Eqs. 2.37 and 2.38 constitute a system in the variables E0 and B0 , which we rewrite in matrix form as 

E0





M11 M12



E0



   = 0. M  =  B0 M21 M22 B0

The matrix elements Mij are given by   ′ −Im (ka) εr Im (ka) − Γm M11 = kz k κ   ω −1 εr M12 = im − 2 Im (ka) a k2 κ   kz −1 1 M21 = im − 2 Im (ka) a k2 κ   ′ Im (ka) Im (ka) , Πm + M22 = ω κ k

33

(2.39)

(2.40) (2.41) (2.42) (2.43)

where the quantities Γm and Πm have been introduced for convenience and are defined as Γm ≡

αm ′ {Jm (κb)Ym′ (κa) − Ym (κb)Jm (κa)} Im (ka)

′ Jm (κb)Ym′ (κa) − Ym (κb)Jm (κa) = Ym (κa)Jm (κb) − Ym (κb)Jm (κa)

Πm ≡ =

(2.44)

ηm ′ ′ {Jm (κb)Ym′ (κa) − Ym′ (κb)Jm (κa)} Im (ka) ′ ′ Jm (κb)Ym′ (κa) − Ym′ (κb)Jm (κa) . ′ ′ Ym (κa)Jm (κb) − Ym (κb)Jm (κa)

(2.45)

In order for our mathematical model to be internally consistent, the determinant of the matrix M must be zero. The reason for this is the following. Because we have not specified the source of the EM fields excited in the DLW structure, the field amplitude E0 cannot be determined. Eq. 2.39 is a homogenous system, and therefore has a unique solution only if its determinant is non-zero; otherwise, the system has an infinite number of solutions, a situation corresponding to the arbitrariness of the radiation source. We conclude that we must require the determinant of M to vanish. By applying this condition and inserting the explicit forms of the matrix elements Mij , we will generate a transcendental equation representing the dispersion relation of the modes supported by the DLW structure. We begin by writing the determinant det M = M11 M22 − M12 M21 = 0. In explicit form, we have  ′   ′ −Im (ka) εr Im (ka) Im (ka) Im (ka) 0 = ωkz − Γm Πm + k κ κ k    −1 εr −1 1 2 2 ωkz − 2 − 2 Im (ka), +m 2 a k2 κ k2 κ

34

(2.46)

(2.47)

which is easily rewritten in a final form:    2 m2 εr kmn 1 1 1+ 2 + 2 2 a2 κmn kmn κmn  ′   ′ Im (kmn a) kmn Im (kmn a) εr kmn + Γmn + Πmn . = Im (kmn a) κmn Im (kmn a) κmn

(2.48)

This is a transcendental equation that must be solved numerically. For each value of m there is a discrete sequence of wavenumber values, enumerated by the index n = 1, 2, 3, ..., for which the equation is satisfied and the corresponding modes are allowed to propagate in the structure. Thus each combination of (m, n) values refers to a mode configuration. In order to solve Eq. 2.48, it is necessary to eliminate either one of the transverse wavenumbers (kmn , κmn ) by writing it in terms of the other. We can do this by recognizing the fact that we are ultimately interested in fields excited by a charge moving at speed vb = βc, and all such modes will have an equal phase velocity of vφ = vb = βc [46]. The angular frequency of a guided mode is then given by ω = vφ kz = βckz , and we can make use of Eq. 2.10 to relate the two transverse wavenumbers as κmn kmn = p . γ εr β 2 − 1

(2.49)

Upon making this substitution and solving the dispersion relation for κmn , the guided mode frequencies will be given by fmn =

1 βc κmn ωmn p . = 2π 2π εr β 2 − 1

(2.50)

It is seen that the frequency is fairly insensitive to the driving beam energy, due to the asymptotic behavior of the beam velocity β in the ultrarelativistic limit.

35

2.1.1.3

Monopole Modes

We can gain some conceptual insight by examining the special case of complete azimuthal symmetry, denoted mathematically by m = 0. Setting m to zero in Eq. 2.48 yields a simplified form of the dispersion relation:    I1 (k0n a) k0n I1 (k0n a) εr k0n + Γ0n + Π0n = 0. I0 (k0n a) κ0n I0 (k0n a) κ0n

(2.51)

It is immediately apparent that this equation will be satisfied if either of the bracketed expressions vanishes; thus we can write two separate dispersion relations I1 (k0n a) εr k0n J0 (κ0n b)Y1 (κ0n a) − Y0 (κ0n b)J1 (κ0n a) = I0 (k0n a) κ0n Y0 (κ0n a)J0 (κ0n b) − Y0 (κ0n b)J0 (κ0n a)

(2.52)

k0n Y1 (κ0n b)J1 (κ0n a) − J1 (κ0n b)Y1 (κ0n a) I1 (k0n a) = , I0 (k0n a) κ0n Y1 (κ0n b)J0 (κ0n a) − Y0 (κ0n a)J1 (κ0n b)

(2.53)

where Γ0n and Π0n have been written out explicitly. This indicates that the modes separate into two classes, namely the well-known [15] TM and TE modes. We will utilize the method described above to solve the TM and TE mode dispersion relations for the specific case γ = 20, εr = 3.8, a = 250 µm, b = 350 µm and obtain mode frequencies using Eq. 2.50. These are the parameters that will be used in the experimental component of this dissertation, and they are chosen for reasons that are discussed later. A clear graphical illustration of the solution to a transcendental equation is given by plotting each side of the equation against the variable κ; the points at which the curves intersect are the solutions κ0n , as shown in Fig. 2.2. The approximate mode frequencies of the first three TM and TE modes, calculated for the given parameters, are listed in Table 2.1. We see that they are within the THz range for this sub-mm dimensional regime.

36

TM modes

0

0

Figure 2.2

20

к

40 ( 1000 m -1)

TE modes

60

0

80

0

20

к

40 ( 1000 m -1)

60

80

Example of graphical solution to transcendental dispersion relation

for TM and TE modes.

Table 2.1

Calculated mode frequencies for first 3 TM, TE, and HEM modes.

Parameters used: γ = 20, εr = 3.8, a = 250 µm, b = 350 µm.

n

TM0n

TE0n

HEM1n

1

368 GHz

542 GHz

356 GHz

2

1.09 THz

1.37 THz

534 GHz

3

1.91 THz

2.25 THz

1.16 THz

37

2.1.1.4

Dipole Modes

For the general case of m 6= 0, there is no separation of modes into pure TM and TE types. In this hybrid electromagnetic (HEM) mode case, the dispersion relation becomes much more cumbersome; however, once an expression is obtained, the solution by numerical methods proceeds in the same manner. It is important to study this case, because it is realistic to expect azimuthal symmetry to be broken in an experimental situation where an electron beam driving the modes is most likely off-axis by some amount. We consider here only the dipole mode case, for which m = 1. Higher-order modes decrease greatly in strength of coupling to a driving electron beam due to the coherence threshold, so they are not as important to consider in this experimental study. Setting m = 1 in Eq. 2.48, we obtain the expression    2 1 εr k1n 1 1 + 2 1+ 2 2 a2 κ1n k1n κ1n    1 I0 (k1n a) + I2 (k1n a) εr k1n 1 I0 (k1n a) + I2 (k1n a) k1n = + Γ1n + Π1n , 2 I1 (k1n a) κ1n 2 I1 (k1n a) κ1n (2.54) where the quantities Γ1n and Π1n are given explicitly as Γ1n =

1 J1 (κb)[Y0 (κa) − Y2 (κa)] − Y1 (κb)[J0 (κa) − J2 (κa)] 2 Y1 (κa)J1 (κb) − Y1 (κb)J1 (κa)

Π1n =

1 [J0 (κb) − J2 (κb)][Y0 (κa) − Y2 (κa)] − [Y0 (κb) − Y2 (κb)][J0 (κa) − J2 (κa)] . 2 Y1 (κa)[J0 (κb) − J2 (κb)] − [Y0 (κb) − Y2 (κb)]J1 (κa)

(2.55)

(2.56)

The subscript 1n on the wavenumber κ has been momentarily suppressed for formatting purposes. Eq. 2.54 is analogous to Eq. 2.51 for the m = 0 modes. Numerical solution of this equation will yield the HEM1n mode frequencies, as

38

HEM modes 0

0

10

20

к Figure 2.3

30 40 ( 1000 m -1)

50

Example of graphical solution to transcendental dispersion relation

for HEM1n modes.

illustrated in Fig. 2.3. The calculated values of the first three HEM modes are included in Table 2.1. It is possible for modes from some or all of these three classes to be excited simultaneously in the DLW structure, depending on the nature of the driving source. Strictly speaking, in the experiment under consideration where an electron beam is the driving source, all three types of modes will be excited due to the inhomogenous nature of a real beam charge distribution. However, the dipole modes are difficult to excite compared to the monopole modes because of their zero coupling field on-axis, a point that will be elaborated in the following section. This fact, paired with the coherent nature of the excitation, will ensure that the lowest-frequency monopole mode will dominate.

2.1.1.5

Transverse Mode Structure

Utilizing the field solutions found in Section 2.1.1.1 and the eigenfrequencies found in the previous section, we can plot the field components against the radial coordinate to gain a more intuitive picture of the transverse structure of the

39

modes. Setting Bz = 0 in Eqs. 2.29 yields the transverse field components for the TM modes; likewise setting Ez = 0 yields the TE mode fields. The nonzero components are shown in Figs. 2.4 and 2.5. The radial variation of the fields is seen to be roughly linear in the vacuum region and oscillatory within the dielectric. Of particular interest is the relative strength of Ez between the TM01 and HEM11 modes. Because these modes are fairly close in frequency (Table 2.1) and therefore subject to roughly the same coherence threshold, it is questionable whether it will be possible to distinguish them in an experimental situation when only the frequency is being measured. The driving electron beam gives energy to a radiation mode when it is decelerated by the longitudinal electric field, which is a maximum on the axis in the TM01 mode case. As shown in Fig. 2.6, HEM11 Ez has a null at r = 0 and grows away from the axis, and the maximum field magnitude is much smaller. Furthermore, the HEM11 field has a dipole asymmetry and so will only couple efficiently to an off-axis beam. Thus we conclude that the TM01 mode should be preferentially excited by a beam that is roughly azimuthally symmetric and on-axis. Fig. 2.7 shows the longitudinal Poynting flux Sz =

1 E 2


× H∗ ·ˆ z, representing

the power flow along the structure, and the electromagnetic energy density uEM = 1 2

(ε|E|2 + µ|H|2) of the TM01 mode as functions of r. The characteristic “donut”

hollow intensity pattern is apparent from the on-axis null in Sz seen in Fig. 2.7a; the discontinuity at the vacuum/dielectric interface, a manifestation of refraction, appears because we are viewing only the longitudinal vector component of S. It is interesting to note that the energy is largely concentrated in the dielectric region, as seen in Fig. 2.7b.

40

TM01 mode: Ez

E0

0

Bθ

Er

E0

ε0 εr ω E0 kz

0

0

0

radius r

a

b

0

radius r

a

b

0

radius r

a

b

TM02 mode: Ez

Er

Bθ

2E 0

2 ε0 εr ω E 0 kz

E0

0

0

0

0

radius r

a

b

0

radius r

a

b

0

radius r

a

b

TM03 mode: Ez

4E 0

Er

4E 0

0

0

0

0

radius r

Figure 2.4

a

b

Bθ

4 ε0 εr ω E 0 kz

0

radius r

a

b

0

radius r

a

Calculated radial field profiles of first three TM0n modes.

41

b

TE 01 mode: Bz

Eθ

Br

B0

B0

0

0

0 - kz B ω 0

0

radius r

a

b

0

radius r

a

b

0

radius r

a

b

TE 02 mode: Bz

Br

5B 0

5B 0

0

0

Eθ 0 -5 kz B ω 0

0

radius r

a

b

0

radius r

a

b

0

radius r

a

b

TE 03 mode: Bz

Eθ

Br

5B 0

5B 0

0

0

0 -5 kz B ω 0

0

radius r

Figure 2.5

a

b

0

radius r

a

b

0

radius r

a

Calculated radial field profiles of first three TE0n modes.

42

b

TM01 E z

HEM11 E z

E0

~E 0 20

0

0

0

Figure 2.6

radius r

a

b

0

radius r

a

b

Comparison of calculated radial Ez profiles of TM01 and HEM11

modes.

TM01 S z

TM01 u EM

(b)

( W • m-2 )

( J • m-3 )

(a)

0 0

Figure 2.7

radius r

a

0

b

0

radius r

a

b

Radial profiles of (a) Poynting flux Sz and (b) energy density uEM

for the TM01 mode.

43

2.1.2

Beam-Driven Wakefields

Our analysis up to this point has been concerned with the eigenmodes in a cylindrical DLW structure, with the source of the fields being left arbitrary. In this section, we consider the response of the structure to a single electron traveling along the axis. Instead of inserting the appropriate source terms in Maxwell’s equations and solving the resulting inhomogenous wave equation, we will make use of an energy-balance technique used by Tremaine and Rosenzweig in their analysis of a slab-geometry dielectric wakefield accelerator [103]. The singleparticle response will then be used to find the complete beam-driven wakefield. vgt

e-

z

1 2 vbt = βct Figure 2.8

Illustration of the role of group velocity in the radiation pulse length.

First, we examine the physical picture. Consider an electron with charge e traveling at a relativistic speed vb = βc along the axis of the DLW. After a time t it has covered a distance vb t = βct, leaving a Cherenkov wakefield behind it. The EM field fills only a part of the volume behind the particle, because as radiation is emitted it moves forward at the group velocity vg . In the same time t, the initial wakefield (now the “tail” of the wake) has moved forward a distance vg t, vacating the volume at the entrance of the structure. This is illustrated in Fig. 2.8. In the figure, the total volume behind the particle (the sum of regions 1 and

44

2) would be filled with electromagnetic energy in the limit of zero group velocity; because the wakefield moves at vg , the volume of region 1 is excluded, leaving region 2 occupied by the radiation fields. Mathematically, the electromagnetic energy left behind the particle (i.e. stored in region 2) in a given mode with fields (E, H) is given by the integrated, timeaveraged energy density Z Z Z Z Z 2π Z βct
1 b huEM i dV = r dr dθ dz ε|E|2 + µ|H|2 4 0 V 0 0

(2.57)

less the amount of energy that has receded from the beginning of the structure along the +z direction Z Z Z

V

1 1 hSz i dV = c c

Z

b

r dr

0

Z

0

2π

dθ

Z

0

βct

dz S · ˆz.

(2.58)

By conservation of energy, this must be equal to the amount of energy that the particle has lost to its interaction with the decelerating field because the deceleration is the only input of energy to the system. According to the wakefield theorem discussed in Section 1.2, the electron experiences a decelerating field Ez,dec = E0 /2 where E0 is the on-axis amplitude of the wakefield driven in the structure; we consider only a single-mode, TM01 wakefield. The energy change of the particle per unit length is then eEz,dec . We equate this to the energy quantified by Eqs. 2.57 and 2.58, yielding  Z b Z 2π  eE0 1 eEz,dec = = huEM i − hSz i r dr dθ. 2 c 0 0

(2.59)

By computing the RHS of this equation using the fields found in Section 2.1.1.1, we can solve for the unknown amplitude E0 and thus fully determine the mode fields excited by the electron. Inserting the explicit form of the TM01 Ez field,

45

the integrated Poynting flux, which represents the radiated power flow, is Z Z Z b πkz µεω ′ ′ hSz i dS = P = RE (r)RE (r)∗ r dr 2 − k 2 )2 (µεω S 0 z =

E02

πkz µεω k4

Z

a

0

I12 (kr) r dr

E02

+

πkz µεω 2 α0 κ4

Z

a

b



Y02 (κb) J12 (κr)

 − 2Y0 (κb) J0 (κb) Y1 (κr) J1(κr) + J02 (κb) Y12 (κr) r dr,

(2.60)

where the mode index subscripts on the wavenumbers have been omitted. The integrated EM energy density, which represents the stored field energy per unit length, is UEM =

µ3 ε2 ω 2 + (µεω 2 − kz2 )

Z b

εkz2 ′ R′ (r)RE (r)∗ + εRE (r)RE (r)∗ 2 − k 2 )2 E (µεω 0 z  ′ (2.61) R′ (r)RE (r)∗ r dr. 2 E

π huEM i dS = 2 S

Z Z

As the algebra involved in manipulating these expressions becomes increasingly long and tedious without leading to a simplified expression, we will be satisfied with calculating the numerical value of E0 for a given structure geometry and charge e. We find, for a = 250 µm and b = 350 µm (400 µm), a value of E0 ≈ 0.04 V/m (0.03 V/m).

2.1.2.1

Group Velocity and Pulse Length

The group velocity vg of the wakefield, representing the speed of energy flow along the waveguide, is simply the constant of proportionality between the total power flow and the field energy per unit length vg =

P . UEM

(2.62)

Evaluating Eq. 2.62 numerically, we obtain vg ≈ 0.37 c (0.29c) for the b = 350 µm (400 µm) tube.

46

Table 2.2 Calculated values of E0 , U , vg , and tpulse for two tube sizes.

b = 350 µm

b = 400 µm

E0

0.04 V/m

0.03 V/m

U01

10.3 µJ

15 µJ

vg

0.37 c

0.29c

tpulse

58 ps

84 ps

Knowledge of the group velocity allows us to estimate the radiation pulse length, referring again to Fig. 2.8. When the driving beam has traveled a length ℓ = vb t = βct along the structure, the initially emitted radiation has traveled vg t = vg ℓ/βc. Thus the physical length of the CCR pulse in the structure is the difference of these, ℓ(1 − vg /βc), the length of region 2 in the figure. For the total tube length L, the radiation exits the structure in a time   L vg L L tpulse = 1− = − , vg βc vg vb

(2.63)

which is the pulse length we should observe in our measurements. For L = 1 cm and vg = 0.37 c (0.29c), we estimate tpulse ≈ 58 ps (84 ps). 2.1.2.2

Coherent Excitation of Wakefields

We represent the energy radiated by a single electron as the total power flow multiplied by the pulse duration: Ue = P · tpulse .

(2.64)

When the driving source is an extended beam charge distribution, wakefields can be driven coherently in the manner discussed in Section 1.1.2.1. We make use of

47

Total Radiated Energy vs. σz

60

b = 350 μm b = 400 μm

U ( μJ)

50 40 30 20 10 0 0

50

100

150

200

250

300

350

σz ( μm ) Figure 2.9

Calculated total radiated energy as a function of σz .

the coherent form factor F (λ) to estimate the energy U radiated by a Gaussian distribution containing N electrons, given by U = Ue · N 2 · F (λ) = Ue · N 2 · e−4π

2 σ 2 /λ2 z

.

(2.65)

Fig. 2.9 shows the dependence of U on σz , calculated for the TM01 mode of the specific structures under consideration, illustrating the effect of coherence. As these curves assume that only a single mode is excited, they are not valid for lower values of σz at which higher-frequency modes would be excited. The bunch length of the beam used in the experiment (Chapter 4) is measured to be σz = 165 ± 15 µm (Section A.2.2). Thus we obtain U ≈ 10.3 µJ (15 µJ) for the b = 350 µm (400 µm) tube. It is seen that, for the given parameters, the experiment will operate somewhat on the edge of the coherence threshold. We can compare these numbers to the total energy lost by the beam as it experiences the integrated effect of the TM01 wakefield along the length of the structure, making use of the wakefield analysis formalism outlined in Section 1.2. We begin by writing the wake potential due to a single particle using Eq. 1.18

48

and subsequently applying Eqs. 1.19 and 1.20 to obtain   Z Z s Z L −1 ∞ z + s − s′ ′ Ulost = ρ(s′ ) ρ(s), ds ds dz Ez,mn z, r = 0, t = e −∞ c −∞ 0 (2.66) where L is the tube length, ρ(s) is the beam charge density function and s is the axial coordinate as defined in Fig. 1.6. Numerical evaluation of this integral proceeds by inserting the appropriate Ez,mn field function (Section 2.1.1.1) and including the E0 amplitude value calculated in Section 2.1.2 above. Using an ideal Gaussian beam distribution ρ(s) =

Q 2 2 √ e−s /2σz σz 2π

(2.67)

with Q = 200 pC and σz = 165 µm, we obtain a value Ulost ≈ 11.7 µJ (15.8 µJ) for the b = 350 µm (400 µm) tube. Fig. 2.10 shows the calculated electric potential difference V (s) (Eq. 1.19) experienced along the structure by a test particle traveling at a position s inside the driving beam. It is seen that some particles at the tail of the beam gain energy as the decelerating electric field oscillates and changes sign.

2.2

Computer Simulation

The commercial particle-in-cell (PIC) electromagnetic simulation code oopic pro [104] was used to simulate the scenario of an ideal Gaussian electron beam traveling along the axis of a DLW structure. The code is 2D, and can simulate a 3D geometry only by the assumption of symmetry. Azimuthal symmetry around an axis was used in this study, corresponding to a 3D cylindrical geometry. The goal of the computational study was to confirm the results found by the analytical method presented in this chapter, and to choose structure dimensions that produce the desired sinusoidal wakefield when excited by a beam such as the one

49

b = 350 μm 200

(a) charge density ( a.u. )

potential difference ( kV )

300

100 0 -100 V (s) ρ (s)

-200 -300

-0.5

0

0.5

1

0.5

1

s ( mm )

b = 400 μm 200

(b) charge density ( a.u. )

potential difference ( kV )

300

100 0 -100 -200

V (s) ρ (s)

-300 -0.5

0 s ( mm )

Figure 2.10

Total electric potential difference V (s), calculated analytically, ex-

perienced by electrons inside a Gaussian driving beam for (a) b = 350 µm and (b) b = 400 µm.

50

available to us at the UCLA Neptune advanced accelerator laboratory. The simulation also offers information on transient behavior and includes any contribution of TM0n harmonics, effects that have been neglected in our analytical treatment. We will see that the results validate the single-mode assumption that we have carried throughout the analysis. Parametric scans, in which the adjustable parameters were varied one at a time and the simulated outcome observed, were used to narrow down the parameter ranges of interest. The final design values of the parameters were then selected from these ranges. Three separate parametric scans were conducted: 1. Hold the inner tube radius constant at a = 250 µm while varying the outer radius b. 2. Hold the wall thickness constant at (b − a) = 100 µm while varying the inner radius. 3. Hold tube dimensions constant at a = 250 µm / b = 350 µm while varying the electron bunch length σz . The goal of the first two scans is to examine the behavior of the excited mode structure as the tube dimensions change. The goal of the third scan is to explore the effect of the length of the driving beam on the coherence of the radiation process.

2.2.1

Case 1: Constant Inner Radius

The longitudinal field Ez on the axis of the structure is shown for varying values of b in Fig. 2.11. For the case of b = 350 µm the field is almost perfectly sinusoidal, indicating that a single mode frequency is excited in the structure. Field plots for successively larger b values clearly show the excitation of additional modes

51

as the wall thickness increases to accommodate longer wavelengths. The power spectra calculated from these field plots are shown in Fig. 2.12. The scaling of the simulated TM0n mode frequencies with b is summarized in Fig. 2.13, including comparison with the analytical calculation. It should be noted that the field plots contain numerical artifacts manifested as small high-frequency oscillations.

2.2.2

Case 2: Constant Wall Thickness

Fig. 2.14 shows Ez along the axis of the DLW tube for varying values of a, holding the wall thickness (b − a) constant. It is interesting to note that although the overall size of the waveguide is increasing, the wakefield excitation is still essentially single-mode because the wall thickness is not increasing. Fig. 2.15 shows the magnitude of the decelerating electric field Ez,dec seen by the driving electron bunch as a function of a. It is seen that the decelerating field, and thus the energy lost by the bunch, scales inversely with a.

2.2.3

Case 3: Varying Bunch Length

The axial Ez field profile is shown for varying values of σz in Fig. 2.16. The loss of coherence is clearly seen as the bunch length grows beyond the value set by the roll-off wavelength λ ≈ 2πσz . At lower values of σz , the excitation of an additional mode appears as the bunch becomes short enough to coherently excite higher frequencies. The corresponding power spectra, seen in Fig. 2.17, show the emergence of a peak at higher frequency corresponding to the higher-order mode. This illustrates the importance of the bunch length parameter to the experiment; σz must be short enough to coherently excite the TM01 mode, but not so short that it excites higher harmonics so as to preserve single-mode operation.

52

On-axis Longitudinal field Ez 20 0

b = 1400 μm

-20 20 0

b = 700 μm

-20 20 0

e-

b = 350 μm

-20

( MV/m )

20 0

b = 275 μm

-20 0

Figure 2.11

Axial Position z (mm)

10

oopic simulation of longitudinal field Ez on-axis for increasing outer

radius b.

53

Simulated Power Spectrum

b = 1400 μm

THz

0.0

0.5

b = 700 μm

b = 350 μm

( a.u. )

b = 275 μm

0.0

0

0.5

THz

0.5

1.0

1.5

ƒ ( THz ) Figure 2.12

oopic simulation of power spectra for increasing outer radius b,

calculated from fields shown in Fig. 2.11. Vertical scale is constant between plots.

54

TM mode frequency 1.0

TM 02

OOPIC analytical

TM 03

ƒ ( THz )

0.8 0.6

TM 01

0.4 0.2 0 300

400

500

600

700

b ( μm ) Figure 2.13

Comparison of analytical and oopic simulation results showing

scaling of TM0n frequency with b.

2.2.4

Beam Energy Loss

An alternative method for estimating the radiated energy is to observe in simulation the energy lost by the drive bunch when it has traveled the length of the tube. The longitudinal phase space of the beam electrons is tracked through the length of the structure, and a histogram of the particle energies in the initial and final distributions is compared. The difference between the statistical mode of energy values at these two points is taken to be the amount of energy ∆E lost by the beam particles. The total amount of energy lost by the beam is then roughly U = N∆E, which serves as an estimate of the radiated energy. The estimate is rough, as the final energy distribution is wide, and some of the particles are actually accelerated rather than decelerated. Figure 2.18 shows longitudinal phase space distributions from the beginning and end of the structure superimposed on the longitudinal wakefield Ez . We find U ≈ 12 µJ (18 µJ) for the b = 350 µm (400 µm) tube. Table 2.3 compares the radiated energy estimates of the three

55

On-axis Longitudinal field Ez 20

a = 500 μm

0 -20 20

a = 400 μm

0 -20 20

a = 250 μm

0

e-

-20

( MV/m )

20

a = 100 μm

0

-20 0

Figure 2.14

Axial Position z (mm)

10

oopic simulation of longitudinal field Ez on-axis for increasing inner

radius a.

56

Decelerating Field 14 OOPIC 1 / a fit

E z,dec ( MV/m )

12 10 8 6 4 2 100

200

300

400

500

a ( μm ) Figure 2.15

oopic simulation of longitudinal decelerating field Ez,dec seen by

driving beam for increasing outer radius a. Fit curve shows the magnitude is inversely proportional to a.

different methods used in this chapter.

Table 2.3

Comparison of radiated energy estimates obtained from three methods.

350 µm

400 µm

power flow, analytical (µJ)

10.3

15

energy loss, analytical

11.7

15.8

12

18

energy loss, oopic

The simulation also illustrates the dramatic effect of the wakefield interaction on the longitudinal phase space of the beam. We see that the full energy spread increases by several times along the structure, which is expected since Ez varies throughout the beam. The driving bunch is essentially “spent” as it exits the DLW, in terms of its quality for subsequent use downstream. This fact compli-

57

On-axis Longitudinal field Ez 20 0

σz = 500 μm

-20 20 0

σz = 300 μm

-20 20 0

e-

σz = 200 μm

-20

( MV/m )

60 0

σz = 10 μm

-60 0

Figure 2.16

Axial Position z (mm)

10

OOPIC simulation of longitudinal field Ez on-axis for increasing

bunch length σz .

58

Simulated Power Spectrum

σz = 500 μm

0.0

0.5

THz

σz = 300 μm

0.0

0.5

THz

σz = 200 μm

( a.u. )

σz = 10 μm

0

0.5

1.0

1.5

ƒ ( THz ) Figure 2.17

oopic simulation of power spectra for increasing bunch length σz ,

calculated from fields shown in Fig. 2.16.

59

b = 400 μm final

Energy ( MeV )

E z ( MV/m )

10.75

(a)

20 10 0

initial

-10

.09 MeV

-20 10.30

8

9

z ( mm )

10

( counts )

11

b = 350 μm final

Energy ( MeV )

E z ( MV/m )

10.75

(b)

20 10 0

initial

-10 -20

10.30

8

Figure 2.18

9

z ( mm )

10

.06 MeV

( counts )

11

oopic simulation of longitudinal phase space of beam electrons for

(a) b = 400 µm and (b) b = 350 µm tubes, with accompanying energy histograms.

cates the incorporation of a beam-driven wakefield device in, e.g., an FEL seeding scenario.

2.3

Chapter Summary

Table 2.4 collects the findings of this chapter that are most relevant in subsequent chapters. These predictions, for two different tube sizes, will be compared to experimental results presented later in this dissertation.

60

Table 2.4 Summary of predictions in Chapter 2.

Tube 1

Tube 2

inner radius

a

250 µm

250 µm

outer radius

b

350 µm

400 µm

TM01 frequency

f

368 GHz

273 GHz

pulse duration

tpulse

58 ps

84 ps

radiated energy

U

11.7 µJ

15.8 µJ

61

CHAPTER

3

Description of Experiment

The goal of this study is to explore the use of a DLW structure as a means of producing narrow-band THz radiation in the form of beam-driven coherent Cherenkov radiation wakefields. The experiment is designed to be a proof-of-principle demonstration of the effectiveness of this technique, which relies on coherent excitation to exclude discrete waveguide modes with wavelengths shorter than the driving electron bunch length. This allows an inherently broadband beam current profile to radiate power into a single frequency, which is selectable by appropriate choice of design parameters. The method is distinct from previously proven Cherenkov FEL scenarios in that it does not employ a gain process, instead relying solely on superradiant (coherent) emission from the ultrashort beam as discussed in Chapter 2. The principal diagnostic employed in the measurements is a standard Michelson interferometer optimized for use at THz wavelengths, used in conjunction with a Si bolometer detector. By using this device to record an autocorrelation interferogram of the emitted radiation, a power spectrum is measured which can be compared to predictions. The expected result is a narrow spectral line at the TM01 mode frequency, with the central frequency shifting when a tube of different

62

dimensions is used. The experimental setup is designed to focus the compressed electron beam provided by the UCLA Neptune accelerator through a DLW and then measure the emitted CCR, minimizing any background radiation polluting the signal. This chapter describes the accelerator system used to carry out the measurements, as well as the design goals, defining constraints, and decisions that became the final form of the experiment. The layout of the hardware used in the measurements is presented, along with details of the design and fabrication of specific pieces. Finally, the operating principles of the primary experimental diagnostics and measurement apparatus are discussed. Results and analysis of the measurements are the topic of Chapter 4.

3.1

The Neptune Accelerator

The design of every experiment is influenced and often dictated by the facility that will accommodate it. All of the experimental work described in this dissertation took place in the Neptune advanced accelerator research laboratory, housed on UCLA campus in Boelter Hall rooms 1000 and 2000. The facility is a collaboration between the Particle Beam Physics Laboratory in the Department of Physics and Astronomy and the Laser-Plasma Group in the Electrical Engineering Department. The source of the electron beam is an RF photoelectron gun, or “photoinjector,” in the SLAC/BNL/UCLA 1.6-cell S-band style [105]. While the present design of photoinjectors has evolved somewhat beyond the version in use at Neptune, this device still represents the present state-of-the-art in high-brightness electron beam sources [106]. The photoinjector is thus named because electrons

63

are “injected” into a copper cavity by a laser pulse striking a cathode plate; they are then accelerated away from the cathode by the RF field resonating in the cavity. The gun is surrounded by a solenoid magnet for the purpose of beam emittance compensation1 . The focusing magnetic fringe fields act to remove phase space correlations that arise as the space-charge-dominated beam undergoes plasma oscillation at the beginning of its violent acceleration cycle [107, 108]. As the beam quickly becomes relativistic, the influence of the space-charge forces is diminished, and proper tuning of the solenoid can place the transverse emittance minimum at the exit of the photoinjector [109, 110]. The energy of the beam as it leaves the gun is approximately 6 MeV, and further acceleration of the beam up to full energy (∼ 12 MeV) is accomplished using a Plane Wave Transformer (PWT) accelerating cavity located downstream. The PWT linac is a standing-wave device, consisting of 9 cavity cells. RF power is provided to the system by a SLAC model XK-5 S-band klystron operating at 2.856 GHz. A comprehensive schematic of the entire accelerator system is shown in Fig. 3.1. The ultimate source of the RF signal is a quartz crystal oscillating at 38.08 MHz. A fraction of this signal is split off for use in a phase-lock loop for the drive laser; the remainder is frequency-multiplied by 75 to serve as a low-level 2.856 GHz seed for amplification in the klystron. The low-level signal is first amplified to ∼ 1 kW by a solid-state amplifier. It is then fed into the klystron buncher cavity, coinciding with the arrival of a ∼ 38.5 kV square pulse at the klystron electron gun. The DC beam generated by the square pulse is microbunched at 2.856 GHz by the input 1 kW signal, and in turn radiates power at this frequency from an output port feeding a waveguide. The power generated by this klystron is roughly 15 – 17 MW, which is split between the photoinjector and linac cavities. The RF power level, and thus the electron beam energy, can be adjusted by 1

See Section 1.1.1 for a brief introduction to beam dynamics formalism.

64

changing the voltage applied to the klystron; however, operation at levels higher than 39 kV is risky, and a similar klystron at Neptune has been driven to failure at around 42 kV. A detailed overview of the drive laser system is shown in Fig. 3.2. The origin of the laser light is a diode-pumped, folded-cavity resonator operating in the infrared at 1064 nm. The laser initially outputs a quasi-CW, 80 MHz train of 10 ps pulses at an average power of ∼ 650 mW. The pulse train seeds a regenerative laser amplifier cavity at a repetition rate that is reduced using a Pockels cell [111], and amplified pulses are expelled from the cavity using a second Pockels cell at a further reduced rate of 5 Hz. A second amplification stage, consisting of a double pass through a lamp-pumped Nd-doped glass rod, outputs ∼ 1 mJ of energy per pulse. The laser light is then frequency-doubled twice using two separate KDP crystals before it shines on the photoinjector cathode at a UV wavelength of 266 nm. This photon energy hν ≈ 4.6 eV is chosen to correspond roughly to the work function of the cathode material, so that electrons are expelled from the cathode into the cavity with a minimum of excess energy (which contributes to thermal emittance). The cathode material in this case is Mg, with a work function of Φ ≈ 3.7 eV; Cu is also commonly used (Φ ≈ 4.6 eV) [112]. A key consideration in the operation of the accelerator is the arrival timing of the drive laser pulse at the cathode relative to the phase of the RF power resonating in the photoinjector cavity. Care must be taken to inject the electrons at the appropriate RF phase in order to form a stable electron beam that is accelerated out of the gun. This relative timing is fine-tuned using a coaxial phase shifter unit that introduces an adjustable phase delay in the low-level RF signal before it is fed to the klystron. Another critical timing relationship is that between the respective phases of the RF power in the gun and linac, which is

65

Figure 3.1

Schematic of entire accelerator system.

66

quartz crystal

38.08 MHz

experimental chamber

ƒ x 75

2.856 GHz

quad magnets

waveguide laser beam coaxial cable

chicane/ spectrometer

38.5 kV

kw amp

HV supply

phase shifter

quad magnets

modulator

klystron

15 - 17 MW

quad magnets

phase shifter

drive laser

linac

solenoid

ephotoinjector

KDP ƒ-doubling crystal

KDP ƒ-doubling crystal

532 nm

266 nm double-pass amplifier

laser cavity

phase-lock

Figure 3.2

regenerative amplifier cavity

1064 nm laser light

vault

photocathode laser room

quartz crystal

Schematic of drive laser system, including regenerative amplifier and

booster amplifier.

controlled by a mechanical phase shifter installed in the waveguide itself; this adjustment determines the RF phase seen by the beam as it arrives in the linac cavity, and allows the energy spread to be minimized or an energy chirp to be imparted for beam compression in a magnetic chicane. The transport of the beam along the accelerator is accomplished using magnets, in the standard fashion. Dipole magnets are used for steering, and quadrupole magnets are used for focusing. In beam dynamics terms, the focusing of a beam corresponds to rotation and subsequent removal of the trace space correlation that is introduced as the beam propagates through a drift. When the correlation is completely removed, the beam is at a focused waist. The final focus for this experiment, which determines the beam spot at the center of the dielectric tube for use in the interaction, is produced using a triplet of permanent magnet quadrupoles (PMQ). Permanent magnets are chosen for their superior achievable field strength over electromagnets. The particular triplet used in these

67

Table 3.1 Nominal beam parameters, before and after compression.

Parameter

Uncompressed

Compressed (at IP)

beam energy

12 MeV

10 MeV

bunch charge Q

200 pC

200 pC

RMS bunch length σz

> 1 mm

165 ± 15 µm

RMS transverse beam size σr

—

80 µm

transverse emittance εx,y

5 mm-mrad

10 mm-mrad

β function

—

1 cm

energy spread ∆E

0.2 MeV

0.7 MeV

measurements was designed and characterized for an inverse Compton scattering experiment previously undertaken at Neptune [113]. The magnetic field gradient in the magnets is ∼ 110 T/m. The smallest focused spot achievable with these magnets for the given beam charge is σr ≈ 40 – 50 µm; in the present study, a spot size of σr ≈ 80 µm was chosen to increase the β-function for easier passage of the beam through the DLW tube.

3.1.1

Magnetic Chicane Compression

Longitudinal compression of the electron beam was necessary to obtain a subpicosecond beam current profile. Compression was accomplished using a standard magnetic chicane compressor system that can also be set to function as a dipole beam energy spectrometer. The Neptune chicane used in this experiment has been studied extensively by Anderson, as described in detail in [13, 114]. The device consists of 4 dipole electromagnets straddling the beam pipe, downstream of the PWT accelerating cavity. As the beam traverses the 4 magnets, it is

68

B⊙

spectrometer mode

⊙

lower energy higher energy

-

e

B Figure 3.3

⊙

⊗

⊗

⊙

compression mode

Longitudinal compression of electron beam by magnetic chicane.

steered outward and again inward, returning to its original trajectory as it exits the chicane. This is illustrated in Fig. 3.3. Beam electrons with higher energies will take a shorter path through the chicane than lower-energy electrons, because their radius of curvature in the magnetic field is larger. This fact is used to compress the beam as follows. If the phase of the RF power input to the PWT is adjusted relative to the injection phase so that the arriving beam sees a steeply sloped part of the RF temporal waveform, the head of the beam (the leading electrons) is accelerated significantly less than the tail and an energy chirp is imparted. By convention, this is called a positive chirp because the energy of the electrons increases in time along the beam. When the strongly chirped beam traverses the chicane, the more energetic tail particles travel a shorter path and catch up to the head particles, in effect compressing the beam charge distribution longitudinally. In the formal context presented in Section 1.1.1, the energy chirp imparted in the PWT cavity is a correlation in the longitudinal phase plane. This correlation is removed as the beam travels through the chicane, in analogy to the action of a drift that removes a transverse trace space correlation when a beam is focused. Continuing the analogy, the role of the drift length is played by the

69

transport matrix element R56 that quantifies the coupling between the longitudinal coordinate and the momentum variation δp/p. The removal of the correlation results in the beam coming to a longitudinal “waist” as it exits the chicane. The Neptune chicane system is proven capable of routinely compressing the electron beam to bunch lengths of σz < 200 µm [13]. As this capability was key to the success of the experiment at hand, the compressed bunch length was measured for verification as described in Section A.2.2 and found to be 165 ± 15 µm. The sensitivity of the bunch compression to the linac phase is illustrated in Fig. 3.4, showing a measurement of coherent radiation emitted as the beam passes a bend magnet downstream of the chicane2 . As seen in the plot, there is essentially no signal detected until a certain phase value is reached; this is the point at which an appreciable energy chirp is imparted on the beam, allowing compression and thus coherent radiation. As the phase progresses past the optimum value, the compression level is reduced and coherence is lost, leading to a drop in the detected signal. This behavior was also observed during the CCR measurements.

3.2

Design of the Experiment

The first step in the design of the experiment was to establish the range of interesting DLW structure parameters, and to choose a subset from that range that is compatible with the Neptune electron beam parameters. After this was accomplished, an experimental layout and hardware were designed to facilitate the collection of data. This section describes the design of the dielectric structures, as well as the experimental setup and diagnostic apparatus. The DLW tube design is based on results from analytical calculations and oopic pro simulations, 2

Presented in more detail in Appendix A.

70

coherent radiation signal vs. linac phase 60

signal ( mV )

50 40 30 20 10 0 85

Figure 3.4

90 95 phase ( deg )

100

Variation of detected coherent radiation signal with changing linac

phase, illustrating effect of energy chirp on compression.

which are presented in Sections 2.1.1.2 and 2.2 respectively.

3.2.1

Dielectric Structures

There are several adjustable parameters with regard to design of the dielectric structures. Namely, they are the length, the inner and outer radii, and the dielectric constant. The ideal tube design for this experiment is one that supports a single THz frequency mode, under the relevant electron beam parameters, and draws a maximum amount of energy from the beam. The beam delivered by the accelerator system imposes perhaps the most rigid design constraints on the experiment. Table 3.1 lists the nominal beam parameters consistently achievable at Neptune, and the compressed-beam parameters existing in this experiment at the interaction point (IP). The IP is defined as the location of the DLW tube center. The beam parameters of key importance to the selection of a dielectric structure are the β-function and the RMS bunch length. The β-function as described in Section 1.1.1, together with the transverse

71

Table 3.2 Specifications of DLW tubes chosen and manufactured for use in final measurements.

inner radius

a

250 µm

outer radius

b

325/350/400 µm

dielectric constant

εr

3.8

tube length

L

1 cm

beam size at the beam waist, dictates the tube length and inner diameter that can be used. This in turn limits the total radiated energy that can be produced. The radius of the tube should be at least a few times σr to allow the beam to pass through without scraping; a value of a = 250 µm was chosen, corresponding to about 3 times the minimum beam size σr ≈ 80 µm. The value of the β-function at the waist, β ∗ , is then calculated to be approximately 1 cm (Eq. 1.6). This corresponds to the distance the beam has traveled from the waist when its RMS √ transverse size σr has grown by 2, informing the choice of a tube length of 1 cm. The bunch length limits the maximum frequency of radiation that can be coherently excited by the beam. To achieve appreciable coherence, σz should be less than the radiation wavelength, as described by the roll-off wavelength condition given in Eq. 1.15 for an ideal Gaussian beam current profile. This restriction informed the choice of inner/outer radii (a, b) combinations of (250,325) µm, (250,350) µm, and (250,400) µm, for which the TM01 mode wavelength is in the vicinity of the coherence roll-off. A summary of the tube parameters used in the final measurements is shown in Table 3.2. The dielectric structures used in the experiment began as custom-pulled hol-

72

low fused silica capillary tubing. The uncoated tubing was cleaved into 10 cm lengths by the manufacturer. To create the outer conducting wall, the tubes were coated with metal using a vapor deposition technique. Two sets of samples were prepared with different coatings; one set received a Au layer ∼ 1 µm thick, and a

second set was coated in a thin (∼ 50 ˚ A) adhesion layer of Cr before being coated with Au. The coated tubes were then cleaved into 1 cm lengths. This final step produced the DLW structures in the form used in the experiment. The primary difficulty encountered during the fabrication of these structures arose during the cleaving. First, it was found that the Au coating flaked off in the absence of the Cr underlayer. The Au-only tubes were abandoned in favor of those with the Cr layer. The presence of this layer should have a negligible effect on the electrical properties of the structure, as its thickness is significantly less than a skin depth in the frequency range of interest. The use of vapor deposition coating to fabricate dielectric-lined waveguide is ideal for preventing the presence of a gap between the outer conductor and dielectric lining, which can have a significant effect on the dispersion relation in the structure and therefore cause problems if the device is to be used, e.g., in a seeded Cherenkov maser experiment [115]. To fabricate a structure with a thinner dielectric wall, one might employ

the inverse of this procedure, coating the inside of a metal tube with dielectric by vapor deposition. Synthetic fused silica was chosen as the material because of its universal availability in fiber and capillary form, which is due mainly to the maturity of the fiber optic industry. While it is not an ideally transparent material in the far infrared spectral region, and indeed is quite absorbing in the THz range, the absorption coefficient is still reasonably low at the frequencies below 1 THz for which this experiment is designed. The absorption coefficient of fused silica at ∼ 375 GHz

73

Figure 3.5

Absorption coefficient and refractive index of fused silica, measured

in the THz spectral range. Reprinted with permission from J. Appl. Phys. 102, 043517 (2007). Copyright 2007, American Institute of Physics.

74

has been measured to be ∼ 0.03 mm−1 by Naftaly and Miles [116], as shown in Fig. 3.5. We can use this value to estimate the total percentage of CCR lost to absorption as the pulse travels through the DLW. The absorption coefficient α is defined in an exponential fashion by T (x) = e−αx ,

(3.1)

where T (x) is the fraction of radiation energy transmitted through a material of thickness x. In our scenario, radiation is emitted at every point along the DLW structure; thus, radiation emitted at the beginning of the tube travels through more dielectric material than does radiation emitted near the end. If we assign the symbol u to represent the amount of energy radiated per unit length of the tube and sum the contributions along the total length L, the total transmission fraction is Ttot

Z L 1 = ue−α(x−L) dx uL 0 Z 1 L α(x−L) e dx = L 0 =

 1  1 − e−αL . αL

(3.2)

Using the values α = 0.03 mm−1 and L = 10 mm, the transmission is approximately 86%, which is an acceptable level. This estimate should be reasonably accurate because the radiation is concentrated within the dielectric wall, as discussed in Section 2.1. The measured refractive index of fused silica is also displayed in Fig. 3.5; the relative permittivity at THz frequencies is therefore εr ≈ 3.8, which is the value assumed throughout this dissertation. This is quite different from the value ∼ 2.1 in the visible range [116].

75

Michelson interferometer

bolometer

dipole magnet

waveguide tapers

compressed electron beam

radiation

horn

dielectric focusing quadrupole tubes triplet

Figure 3.6

3.2.2

recollimating quadrupole triplet

OAP mirror

Schematic of first version of experimental setup.

Experimental Layout

The layout of this experiment was prepared in two versions and ultimately fixed in the form that lent itself best to the final measurements. This section will describe both iterations of the setup. The first version of the layout is diagrammed in Fig. 3.6. The incoming compressed electron bunch is focused strongly by a permanent magnet quadrupole triplet (PMQ) to a spot size σr ∼ 80 µm. The β-function of the beam at this point, while small, is sufficient to allow transmission through the 1 cm tube. Immediately following the tube, the quickly diverging beam enters a second PMQ triplet where it is recollimated and then dumped upward by a permanent magnet dipole (PMD). A Faraday cup or integrating current transformer (ICT) will measure the final beam charge as it exits the top of the dipole. As the CCR produced by the interaction leaves the tube it propagates collinearly with the electron beam, guided by waveguide elements described in Section 3.2.3, continuing straight through the PMD to an off-axis parabolic mirror

76

(OAP) that collimates and directs it 90 degrees to a window in the side of the vacuum chamber. There were two problems encountered with this setup: 1. trace-3d [117] beam envelope simulations indicate that it is possible to propagate the beam through this system while simultaneously maintaining a small symmetric transverse spot at the center of the dielectric tube and a reasonable beam size at the exit of the dipole magnet. However, the large energy spread of the compressed beam imposed a very shallow error tolerance in the magnet settings, making a reliable implementation of this configuration essentially impossible with our system. We were not able to observe a beam spot on a screen at the PMD beam dump, instead seeing only a faint flashing produced by the smeared out beam. 2. The PMD magnet used to dump the beam constitutes a powerful source of coherent radiation that produces a very strong background signal. Measurements of this radiation are the topic of Appendix A. The second version of the setup, diagrammed in Fig. 3.7, is aimed at allowing measurement of the final bunch charge while reducing the level of background radiation. The recollimating PMQ triplet was removed and the horn antenna mounted on a linear stage side-by-side with a Faraday cup. This allowed us to move the Faraday cup directly in line with the dielectric tube to catch the exiting electron beam. The dipole magnet was also removed, allowing the electrons to continue forward and strike the OAP mirror. Coherent transition radiation produced by this beam dumping and coherent diffraction radiation from apertures in the beam path do present a measurable background; the signal-to-noise ratio (SNR) of CCR to CTR/CDR was established at roughly 9:1 by measuring the energy of the radiation emitted when the beam passed through a steel “dummy

77

Michelson interferometer

bolometer

compressed electron beam

radiation

Faraday cup

waveguide tapers

horn focusing quadrupole triplet

Figure 3.7

OAP mirror

dielectric tubes

Schematic of second version of experimental setup.

tube” and comparing it to that of a dielectric tube, as described in Section 4.2.1. Photos of the setup, as implemented in the final measurements, are shown in Figs. 3.8 − 3.10. 3.2.3

Transport of THz Radiation

Transport of the radiation produced is an important experimental consideration. There are two main design constraints that need to be met. First, the experiment is housed in a vacuum chamber that has viewports and access ports in fixed places. This dictated the location of the final OAP mirror, as it must be aligned with the output viewport in the side of the vacuum box. Second, the setup must allow the electron beam to pass through with a minimum impact on the radiation signal we are measuring. A sequence of circular waveguiding elements was placed coaxially with the beam trajectory, allowing the beam to pass through while the radiation continues behind it.

78

Figure 3.8

Experimental setup. Electron beam travels from left to right. Z-cut

quartz window is seen in upper right.

79

tubes

1 cm

Figure 3.9

Closeup of holder containing DLW tubes (center). The focusing PMQ

triplet is seen on the left, and the circular waveguide on the right.

80

Figure 3.10

Experimental setup. The OAP mirror is seen on the right, and the

horn antenna in the center.

horn block

dielectric tube

Figure 3.11

circular waveguide

launching horn

Illustration of circular waveguide elements.

81

Fig. 3.11 shows an exploded view of the waveguide transport line following the dielectric tube. As the emitted radiation exits the tube following the electron beam, a gradual waveguide taper in the form of a conical hole machined in an aluminum block serves to adiabatically match the mode into a second circular waveguide of larger diameter. This waveguide terminates in a conical horn antenna, which directs the radiation forward toward the OAP. Photos of this hardware are shown in Fig. 3.12. The opening angle of the horn is chosen to optimize the antenna gain in the wavelength range of interest. The throat of the horn is taken to be roughly the location of the effective source of waves emanating from its end, and thus it is located at the focal point of the OAP. This creates a collimated quasi-optical beam propagating toward the output window. The window material is Z-cut crystal quartz, chosen for its low absorption in the THz range [118]; the Z-cut characteristic, meaning that the cut places the Z crystal axis normal to the window surface, eliminates the effects of birefringence. A pragmatic approach to the analysis of an overmoded conical horn antenna is presented by Milligan [119]. The gain at wavelength λ of a conical horn is given by Gain = 20 log10



2πR λ



− ATL − PEL,

(3.3)

where R is the horn aperture radius. ATL and PEL stand for amplitude taper loss and phase error loss respectively; values of these quantities for various horn geometries are given in tables in [119]. Based on Eq. 3.3, we choose an opening half-angle of ∼ 0.14 radians. The tapered sections and the transition from the dielectric-lined tube to unlined waveguide represent discontinuities that affect the transmission of radiation through the transport line. While a detailed analysis is not undertaken in this dissertation, we mention here in brief a few of the relevant design considerations.

82

Figure 3.12

Photos of radiation transport hardware.

83

It is reasonable to expect some reflection as the Cherenkov wakefield passes from the end of the DLW structure and into the un-lined tapered section [120]. The influence of this effect is assumed to be minimal in this experiment, as a large discrepancy in the measured power level was not observed. The taper geometry also presents a potential source of impedance mismatch. The primary detrimental effects to consider in a waveguide taper design are reflection and mode conversion. Both processes result in the loss of some of the power in the preferred mode, as energy is transferred to the excitation of parasitic modes and to reflections at a discontinuity in the waveguide wall. These effects are minimized by destructive interference between undesirable mode excitations, which relies on there being an appreciable phase difference between these modes. Therefore, a taper must be long enough to allow phases to differ substantially [121]. Care was taken to ensure that the taper length in this design is greater than the wavelengths of interest by more than a factor of ten. Unconventional machining and fabrication methods were employed to manufacture the radiation transport hardware. The circular waveguide and horn antenna, which are a single piece, were created by a process known as electroforming. An aluminum piece called a mandrel was turned on a lathe to match the desired inner dimensions of the horn piece. The mandrel was then coated with a thin layer of Au and an outer layer of Cu a few mm thick (Fig. 3.13a). Next, the ends were machined off to make them smooth and perpendicular and to make the piece the correct length (Fig. 3.13b). Finally, the aluminum mandrel was etched away, leaving only the copper horn piece. The waveguide taper block was initially fabricated on a conventional milling machine, with small pilot holes drilled in the position of the tapers. The conical taper holes were then cut by electric discharge machining.

84

(a)

(b)

Figure 3.13

Horn antenna during electoforming process, (a) before machining

and (b) after machining.

85

3.2.4

Hardware Design

The dielectric tubes rest in a holder mounted on a remote-controlled 3-axis mover. The holder, pictured in Fig. 3.14 and in the lower left panel of Fig. 3.12, is a monolithic piece of aluminum designed to be a multipurpose probe. In addition to holding the DLW tubes, the probe contains a metal screen used for viewing the transverse beam shape via optical transition radiation (OTR). The probe holds two tubes, which can be switched in and out of the electron beam path by scanning the probe laterally with the 3-axis mover. A further lateral scan can position the probe so that the electrons strike the OTR screen, which is mounted at 45 degrees to the beam axis and viewed by an overhead video camera. The beam strikes the screen at a longitudinal position matching that of the tube center, so that the beam spot viewed here corresponds to the shape and size of the beam at the longitudinal center of the tube. The video pixel size was calibrated using a target printed with lines of known widths, allowing measurement of the beam size from the video image. The beam focusing was tuned to place the waist roughly at this OTR screen position, as viewed on the camera, ensuring that focusing of the beam through the dielectric tube was optimal.

3.3

Measurement Apparatus

This section describes the key measurement apparatus used in the experiment. The primary diagnostic is a standard Michelson interferometer. An infrared radiation detector is used to measure the combined signal output from the interferometer. Two types of IR detectors were used: a Golay cell detector and a Si bolometer. This section discusses the operating principle of each, and compares their specifications and performance. The bolometer, while more sensitive than

86

OTR screen tube grooves

electron beam

Figure 3.14

Multipurpose hardware, functioning as dielectric tube holder and

OTR screen.

the Golay cell, requires cryogenic cooling and is cumbersome to use. The Golay cell was used in measurements for which reduced sensitivity was tolerable, and also when an absolute energy calibration was required3 .

3.3.1

Michelson Interferometer

The Michelson inteferometer allows one to measure, on a multi-shot basis, the autocorrelation interferogram of a radiation pulse, proceeding as follows. As depicted in Fig. 3.15, the incident radiation pulse A enters the interferometer and is divided in two parts by the beam splitter. Pulse B is reflected toward mirror M1, while pulse C continues through to mirror M2. Both pulses are reflected backward by their respective mirrors, where they interfere at the beam splitter and continue on to the OAP mirror and into the detector. The signal recorded by the detector represents the total integrated intensity of the superposition of 3

The bolometer is incompatible with the calibration method used; see Section 3.3.2.3.

87

the two pulses: Umeas (τ ) =

Z

∞

Itot (t) dt =

−∞

Z

∞

−∞

|E(t) + E(t + τ )|2 dt.

(3.4)

τ is the temporal delay between the two pulses, which can be adjusted using a motorized stage to move mirror M1 and thus change the optical path length of pulse B. Multiplying out the integrand in Eq. 3.4, we obtain Z ∞
Umeas (τ ) = |E(t)|2 + 2|E(t)E(t + τ )| + |E(t + τ )|2 dt,

(3.5)

−∞

noting that the middle term Z

∞

2|E(t)E(t + τ )|dt

(3.6)

−∞

is proportional to the autocorrelation of the electric field. The remaining two terms represent the baseline energy level detected when the delay is large enough that the pulses do not overlap. This shows that the field autocorrelation can be measured directly by scanning the optical path delay τ and recording the detected energy at each step. The autocorrelation of a radiation pulse is related in a simple way to the temporal profile of the pulse and therefore contains information about its frequency content. The Wiener-Khinchin theorem [122] states that the autocorrelation of a pulse and its power spectrum form a Fourier transform pair. Thus we can extract the power spectrum of a pulse from the autocorrelation data in a straightforward manner; we will rely on this fact to enable our spectral measurements of the CCR. The autocorrelation scan must be configured appropriately for the radiation being measured. The Nyquist sampling theorem 4 [123] states that a signal containing frequencies up to fmax can be exactly reconstructed if it is sampled at a 4

Also called the Shannon sampling theorem or simply the sampling theorem.

88

OAP

A

M1

B

D

splitter

C M2

bolometer

Figure 3.15

Standard Michelson interferometer.

minimum rate greater than 2fmax . The time step used in recording interferograms in this experiment must therefore be smaller than 1/2fmax, where fmax is taken as the highest frequency expected to be present in the CCR pulse.

3.3.2

Radiation Detection

It is important to understand the basics of infrared sensing devices in order to make an informed choice on which detector to use. Several factors play a role in the sensitivity of a detection device. A key figure of merit is the noise equivalent power, or NEP. The NEP of a detector is defined as the amount of radiation power, normalized to the noise bandwidth ∆fn , that will evoke a response with a SNR of 1 from the detector. Any incident power level less than the NEP, in units of

√W , Hz

will not be observable above the noise level. The noise bandwidth

∆fn ∝ 1/∆t is a measure of the integration time ∆t required for the detector signal to rise appreciably above noise fluctuations.

89

Detectors are often compared using the standard figure of merit specific detectivity, defined as the NEP normalized to the active area A of the detecting element: ∗

D ≡ D ∗ is usually quoted in units of

√

√

A . NEP

Hz·cm , W

(3.7)

and allows direct comparison between

detectors having different areas. This is important because the physical size of the detecting element has a strong effect on the SNR [124]; a detector with a low NEP may also have a low SNR if its area is small. IR detectors are fairly slow, with rise times on the order of milliseconds, and therefore are typically used for measuring time-integrated signals.

3.3.2.1

Golay Cell Detector

The Golay cell is commonly used for THz detection, and is sensitive over roughly 20 GHz – 20 THz [125, 126]. The device, diagrammed in Fig. 3.16, consists of a cell of xenon gas, sealed at one end by a blackened plate and at the other by a reflective membrane. An LED shines on the membrane, and the reflected light is measured by a photocell detector. When radiation is incident on the absorbing black plate, the gas is heated and expands. This expansion causes the reflective membrane to stretch, which changes the amount of reflected light received by the photocell. When the response of the photocell circuit is calibrated, the output voltage can be used as a measure of the amount of energy incident on the absorbing membrane. The NEP (at a modulation frequency of 12.5 Hz) is quoted by the manufacturer to be 10−10 ×109

√

Hz·cm . W

90

√W , Hz

yielding a detectivity D ∗ ≈ 5.3

LED absorber

incident radiation

xenon gas cell

reflective membrane

external circuit photocell Figure 3.16

3.3.2.2

Diagram of Golay cell detector.

Bolometer Detector

A bolometer is another type of thermal IR detector [127], diagrammed in Fig. 3.17. Radiation incident on a thermally sensitive element, which is coupled weakly to a heat sink, changes the resistance of the element. The change in resistance is detected by, e.g., a Wheatstone bridge circuit [128] connected to the element. The high thermal sensitivity of the sensing element requires that the bolometer be operated at very low temperatures, in order to reduce ambient blackbody radiation emanating from nearby objects. Our particular detector is housed in a liquid helium cryostat, and is cooled to a temperature of 4.2 K prior to operation. The detecting element is 2.5 mm square, composed of Si bonded to an absorbing diamond substrate. Incident radiation is collected by a gold-plated Winston cone [129] before falling on the Si element. The NEP is quoted by the manufacturer to be 1.16 × 10−13

√W Hz

[130], which gives D ∗ ≈ 2.1 ×1012

This is approximately 400 times better than that of the Golay cell.

91

√

Hz·cm . W

incident radiation Winston cone absorber thermal resistor

Si

external circuit

thermal link

heat sink

Figure 3.17

3.3.2.3

Diagram of Si bolometer detector.

Detector Calibration

Measurements of the total CCR energy emitted were made possible by the use of a calibrated Golay cell detector. The calibration measurements, performed at UCLA in 2006 [131], made use of a 10.6 µm-wavelength CO2 laser cross-calibrated to a precision calorimeter. The measurement yielded a calibration value of 7.6 mV/µJ. We have taken this value as an upper bound on the sensitivity of the detector, since it may have degraded in the last two years. As any age-induced degradation would surely reduce the voltage response of the device, we dismiss this possibility as unimportant since an inflated conception of the sensitivity would merely cause error on the side of more conservative results. The bolometer detector also used in this experiment has an HDPE5 window covering the input, which fully attenuates radiation below 15 µm, precluding calibration with the CO2 laser. The Golay cell also has an HDPE window that transmits approx5

High-Density Polyethylene

92

imately 40% of 10.6 µm light and 90% of THz light6 , and this attenuation is accounted for in the calibration.

3.4

Chapter Summary

This chapter has described an experimental context for verification of the analytical and computational results of Chapter 2, with measurements to be presented in Chapter 4. Details of the linear electron accelerator at the UCLA Neptune advanced accelerator research laboratory have been presented, and the properties of the beam generated by this system utilized in the experimental design. The chapter discusses details of the fabrication and material properties of the DLW structures, as well as the design and layout of the experimental setup and associated hardware. Finally, operating principles and important aspects of the diagnostic apparatus and radiation detection devices are discussed.

6

According to the product manual.

93

CHAPTER

4

Measurement of Coherent Cherenkov Radiation

The central piece of experimental evidence sought is a measured frequency spectrum of coherent Cherenkov radiation emitted from the end of the dielectric-lined waveguide structure. The spectral information is unfolded from the raw interferogram data via discrete Fourier transform analysis, yielding the frequency content of the radiation pulse for comparison with predictions. The presence of a single spectral line of transform-limited bandwidth would evidence the excitation of a single mode in the structure and prove the operation of the device at the design frequency. Another important piece of information is the amount of CCR energy produced. The viability of this technique as a method for THz power generation is dependent on its ability to produce a copious amount of radiation; therefore, it is essential to measure an emitted energy that is consistent with the impressive predictions. This chapter presents the procedure and results of these measurements in their entirety. First, the setup of the diagnostic apparatus and data collection process is detailed. Next, the results of the measurements in the form of raw data are presented and assembled into plots that summarize the measured quantities. The agreement of the results with analytical calculations and computer simulations

94

is then discussed, and it is seen that the measurements match very well with predictions. Finally, a conclusion is presented summarizing the findings of this study and their implications in the context of THz source research.

4.1

Setup for Measurements

The experimental setup inside the vacuum chamber directs a roughly collimated beam of radiation out of the chamber through a viewport. The window material is Z-cut crystal quartz, chosen for its low absorption at THz frequencies. There are two different configurations of the diagnostic apparatus, for measuring either the total energy in the pulse or the power spectrum of the radiation. Both configurations are mounted somewhat simultaneously on a lightweight optical breadboard adjacent to the chamber, and a linear rail system is used to toggle between them with a minimum of rearrangement required. A switch between setups can be accomplished without significant disruption of electron beam operation. To measure the total energy in the pulse, a 90-degree off-axis parabolic mirror is placed just outside the window as depicted in Fig. 4.1a. The mirror focuses the radiation into a Golay cell detector placed at its focal point. The calibration of this detector is described in Section 3.3.2.3. Positioning of the mirror and detector was initially laid out according to the center of the output window and the manufacturer-specified value of the OAP focal length. Transverse alignment of the detector was then fine-tuned using a HeNe laser aligned to the electron beam trajectory from roughly three meters upstream. For power spectrum measurements, the interferometer device is placed directly adjacent to the experimental chamber, aligned to the output viewport (Fig. 4.1b). This apparatus contains an OAP that focuses the recombined light pulse to an

95

(a) energy measurement setup

vacuum window radiation

Golay cell

(b)

OAP

interferometric setup

vacuum window

radiation

Michelson interferometer

OAP

motion controller bolometer

Figure 4.1

Schematic layout of experimental diagnostics.

96

external point where a detector is placed. A cryogenically-cooled Si bolometer detector was used for this purpose; its improved sensitivity over the Golay cell compensated for the large signal loss due to absorption in the beam splitter inside the interferometer. It was found that the level of CCR emitted was very sensitive to fluctuations in the condition of the electron beam. Most prominently, the RF phase in the linac cavity tends to drift relative to the beam injection timing, which affects both the compression and transport of the beam. The RF phase is normally expected to be constant for 10 – 30 minutes before needing to be re-optimized. Therefore, it was essential to complete an autocorrelation scan in as short a time as possible to ensure that the radiation pulse being observed was consistent for the duration of the multi-shot measurement. To accomplish this, the motorized linear stage used to scan the optical path delay within the interferometer was controlled remotely through a GPIB inteface. The scanning and data acquisition process was then automated using a computer program created in LabView. The program moved the stage in specified increments over a preset range, recording and averaging the detector signal at each point and saving the results to a data file. Despite the automation, a scan would take between 20 and 60 minutes to run depending on its physical length. This made it difficult to measure a complete scan. In some cases several attempts were required to obtain a quality interferogram, and many data sets were tainted by changes in the electron beam conditions part-way through the scan.

97

4.2 4.2.1

Results and Discussion Control Measurement

There is coherent THz radiation produced in any beamline carrying high-charge, subpicosecond electron bunches. Therefore it was necessary to do a control measurement to eliminate the possibility that the purported CCR might actually be THz radiation emitted in other forms as the electron beam passed through upstream magnets and apertures, and to establish the fraction of background noise present in the signal. In a crowded experimental setup, there are several possible sources of this background radiation. Strong dipole magnets used in the chicane compressor may present a source of coherent synchrotron and edge radiation. Various pieces of aluminum hardware placed near the beam trajectory can give rise to wakefields; in particular, the passage of the beam through small apertures such as the PMQ magnet openings or the DLW tube ends produces coherent diffraction radiation, which in the limit of a closed hole is identical to transition radiation [42]. Further, radiation is emitted from transverse deflections of the beam due to steering magnets and focusing quadrupoles [132]. The setup for the control measurement was identical to that for the CCR measurement, except the dielectric tube was replaced by a steel tube of the same dimensions. The steel tube presents an aperture to the electron beam that is the same size as that of the dielectric tube, but will not produce any Cherenkov radiation. This allowed the amount of background radiation to be quantified and compared to the total amount of radiation emitted from the dielectric tube. Care was taken to compare the two detected signals side-by-side; the tubes were placed simultaneously in the two slots of the aluminum tube holder, and the detected signal recorded for each tube in rapid succession by lateral movement of the 3-

98

axis probe. This comparison was performed several times and during different beam runs, re-optimizing the alignment each time, to ensure that the conclusion was not skewed by one of the tubes being aligned less optimally to the beam than the other. It was observed that the signal produced by the dielectric tube was consistently higher than that from the steel tube by a ratio of roughly 9:1. Thus we conclude that a significant amount of CCR is indeed produced in the dielectric tube and is measurable above background sources of long-wavelength coherent radiation. The nominal amount of background radiation present in each measurement is taken to be approximately 10%.

4.2.2

Spectral Measurements

The power spectrum measurement was accomplished by applying standard discrete Fourier transform data analysis techniques [133] to an autocorrelation of the radiation pulse emitted from the tube end. The measurement yields raw data in the form of an interferogram, which is a plot of the total time-integrated intensity at the detector as a function of the temporal delay between split optical paths within the interferometer device. Each plot should therefore have a peak at the zero-delay position, corresponding to complete constructive overlap of the two split pulses, and extend symmetrically on either side. The qualitative structure of an interferogram offers clues to the frequency content of the radiation pulse. For example, a broadband pulse with a Gaussian profile will yield a similarly shaped autocorrelation; this is illustrated mathematically in Fig. 4.2. In contrast, a quasi-sinusoidal pulse composed mostly of a single frequency such as that in Fig. 4.3a will yield an oscillatory autocorrelation like the one in Fig. 4.3b. The latter case is very similar to what we expect to measure. Examples of raw measured interferograms for two tube sizes are shown in

99

Electric Field ( a.u. )

(a)

-10

-5

0 time ( ps )

5

10

5

10

Autocorrelation ( a.u. )

(b)

-10

-5

0 delay ( ps )

An example, calculated analytically, of (a) the electric field of a

Figure 4.2

Gaussian radiation pulse as a function of time and (b) its autocorrelation.

Electric Field

( a.u. )

(a)

-30

-20

0 time ( ps )

-10

10

20

30

Autocorrelation

( a.u. )

(b)

-60

Figure 4.3

-40

-20

0 delay ( ps )

20

40

60

An example, calculated analytically, of (a) the electric field of a quasi-

sinusoidal radiation pulse as a function of time and (b) its autocorrelation. Note the difference in the horizontal scales.

100

Figs. 4.4 and 4.5. It is apparent that these scans are not quite symmetric, and that the peak signal is often not at the zero-delay position. This is due to small changes in the electron beam conditions during the scan, as discussed in Section 4.1 above. The step size used in the scans was chosen according to the Nyquist theorem outlined in Section 3.3.1. The largest time step size used in any of the scans was 0.4 ps, and often a smaller step was used. The largest frequency theoretically observable in our data is thus approximately

1 2(0.4ps)

= 1.25 THz to

2.5 THz. This upper bound is sufficient, as it is well above the range of frequencies expected to be present in the measured signal. The regular periodicity of the unprocessed interferograms indicates that the autocorrelated pulse is largely sinusoidal, an unequivocal signature of the narrowband CCR we are studying. Another telling characteristic is the temporal length of the scans. A broadband CTR pulse would have a temporal profile matching that of the subpicosecond electron beam and thus produce a correspondingly short autocorrelation. These interferograms extend to ∼ 20 – 30 ps on either side of the zero position and do not yet show a dying out of interference fringes; as the length of an autocorrelation is by definition twice that of the measured pulse, we infer that we are observing a radiation pulse that is greater than 30 ps in duration. The longest scan recorded is the one in Fig. 4.5b, which is a half-scan extending from zero delay to 60 ps. This plot shows a reasonable dying off of interference fringes while maintaining a constant baseline value, revealing a pulse length of ∼ 60 ps or longer full duration. This is consistent with the pulse duration tpulse calculated in Section 2.1.2.1. Corresponding power spectra, obtained by calculating the discrete Fourier transforms of the data in Figs. 4.4 and 4.5, are superimposed in Fig. 4.6. The spectra contain features that we can compare directly to analytical and compu-

101

Raw Measured Autocorrelation

(a)

-30

-20

0

-10

10

20

30

(b)

-20

-10

0

10

20

(c)

detected energy ( a.u. )

-20

-10

0

10

20

(d)

-30

Figure 4.4

-20

-10

0 delay ( ps )

10

20

30

Raw measured autocorrelation scans for b = 350 µm tube.

102

Raw Measured Autocorrelation

(a)

-30

-20

0

-10

10

20

(b)

10

20

30

40

50

detected energy ( a.u. )

0

60

(c)

-30

-20

-10

0

10

delay ( ps )

Figure 4.5

Raw measured autocorrelation scans for b = 400 µm tube.

103

Measured Power Spectrum,

(a)

interferogram length 60 ps 60 ps 40 ps

( a.u. ) 0

0.2

0.4

0.6 ƒ ( THz )

Measured Power Spectrum,

(b)

0.8

1.0

b = 400 μm

interferogram length 50 ps 120 ps 50 ps

( a.u. ) 0

b = 350 μm

0.2

0.4

0.6

0.8

1.0

ƒ ( THz )

Figure 4.6

Measured power spectra for (a) b = 350 µm tube and (b) b = 400

µm tube.

104

Table 4.1 Comparison of measured central CCR frequency with analytical and computational predictions.

b = 400 µm measured TM01

b = 350 µm

268 ± 3 GHz 367 ± 3 GHz

b = 325 µm —

oopic TM01

266 GHz

365 GHz

445 GHz

calculated TM01

273 GHz

368 GHz

451 GHz

TM02

764 GHz

1.09 THz

1.40 THz

TM03

1.30 THz

1.91THz

2.51 THz

TM04

1.87 THz

2.77 THz

3.67 THz

TE01

387 GHz

542 GHz

694 GHz

TE02

932 GHz

1.37 THz

1.82 GHz

TE03

1.51 THz

2.25 THz

3.00 THz

TE04

2.10 THz

3.14 THz

4.19 THz

HEM11

248 GHz

356 GHz

458 GHz

HEM12

378 GHz

534 GHz

688 GHz

HEM13

802 GHz

1.16 THz

1.51 THz

HEM14

1.36 THz

1.99 THz

2.61 THz

105

Full Bandwidth Δƒ (GHz)

Measured Bandwidth 80

data 1 / Δt

60 40 20 0 40

Figure 4.7

60 80 100 Autocorrelation Length 2Δt (ps)

120

Measured full bandwidth ∆f vs. autocorrelation scan length 2∆t.

tational predictions made using the model discussed in Chapter 2. Most notable is the TM01 mode resonance, which appears prominently as a dominant narrow peak. The consistency of the central frequency of the peak is clear, albeit with varying degrees of noise present in the different spectra, whose vertical scales have been normalized. Table 4.1 shows the calculated mode frequencies and the central frequencies observed in the measured spectra for the different tube sizes. The measured values quoted here were obtained by averaging the central frequency locations taken from the power spectrum plots for each tube size, and are seen to agree with calculated predictions within . 1%. The corresponding error bars were estimated by the standard deviation of these averaged values. The spectra also demonstrate the expected shift in central frequency when the tube geometry is changed. While structures with three different b values were planned, only two were manufactured because the thinnest-walled tubes were very difficult to cleave without breaking. We compare power spectra of tubes with b = 350 µm and b = 400 µm, and find that the expected ∼ 100 GHz shift is cleanly resolved in the measurements.

106

The bandwidth of the central peak observed in each spectrum, taken as the full baseline width of the peak, is ∆f ≈ 25 – 70 GHz depending on the length of the autocorrelation scan. The majority of the autocorrelation traces measured are partial scans, truncated by either hardware limitations or drastic changes in beam conditions, and therefore the spectra exhibit an artificially high bandwidth. Fig. 4.7 shows a plot of the measured bandwidth as a function of the autocorrelation scan length. Pulse length values are quoted as full widths because standard FWHM or σ values are somewhat ill-defined due to the sinusoidal nature of the radiation pulse. The accompanying solid curve denotes the lower bound on measurable bandwidth set by the time-bandwidth product ∆f ∆t ≈ 1 [134], where the radiation pulse length ∆t is taken to be half the length of the autocorrelation trace. We infer that the bandwidth of the observed spectral peaks is essentially transform-limited, and not an indication of an inherently broadened radiation process.

4.2.3

Radiated Energy Measurements

After repeated careful attempts at optimizing the electron beam transport and compression in order to observe the highest signal possible at the calibrated Golay cell detector, the maximized signal was taken to be a lower bound on the CCR energy emitted from the DLW structure. Nominally the highest signal observed, over multiple data collection runs, was approximately 80 – 90 mV for an incident driving electron bunch charge of ∼ 200 pC. Subtracting the 10% background level (Section 4.2.1) and dividing by the Golay cell calibration factor 7.6 mV/µJ (Section 3.3.2.3), we arrive at a value of ∼ 10 ± 2 µJ (compared to ∼ 2 mJ electron beam energy). Per the > 60 ps pulse duration, this corresponds to a peak radiated power level of ∼ 150 kW. Roughly the same value was measured for

107

both tube sizes; while the analytical calculation and oopic both predict a higher energy radiated in the b = 400 µm tube, due to the nature of this measurement we are satisfied with verifying the approximate level of energy radiated and do not expect to be able to resolve the difference in the level between tubes. As described in Section 3.2.2, a Faraday cup is installed so that it can be positioned directly after the waveguide taper to intercept the beam soon after it exits the dielectric tube. An integrating current transformer is used upstream to measure the beam charge before the tube. This configuration was designed to allow measurement of the amount of beam charge transmitted through the DLW, and thus quantify the amount of charge participating in the Cherenkov wakefield interaction. It was seen that roughly the same amount of charge was measured by the Faraday cup whether or not the tube was aligned properly with the beam; it appears that enough charge leaks through the tube holder apparatus, regardless of its positioning, to wash out a measurement of the transmitted charge. The amount of charge measured by the Faraday cup was roughly 10 – 25% lower than that measured by the upstream ICT. This can serve to roughly quantify the loss of charge in the interaction; we observed that charge loss was greater at higher charge values, as is expected due to the detrimental effect of space charge forces on the focusing of a high-charge beam. For this reason the magnitude of radiated energy measured was seen to vary linearly with the incident beam charge, instead of the quadratic charge dependence characteristic of a coherent radiation process (Section 1.1.2.1).

4.3

Conclusion

This dissertation reports the first direct observation of narrow-band THz coherent Cherenkov radiation driven by a sub-picosecond electron beam in a dielectric

108

Measured Power Spectrum

b = 350 μm b = 400 μm

( a.u. )

700 um OD

800 um OD

0

0.2

0.4

0.6

0.8

1.0

ƒ ( THz )

Figure 4.8

Side-by-side comparison of spectra from two tube sizes.

wakefield structure, representing Cherenkov FEL operation in the superradiant regime. The measured quantities and predictions are summarized in Table 4.2. We evaluate the success of the experiment in two capacities: the agreement of the measured spectral characteristics with predictions, and the agreement of the measured radiated energy level with predictions. With regard to the former, the observed power spectra have a narrow (transform-limited bandwidth) spectral line with central frequency matching that of the predicted (analytical and computational) TM01 mode resonance to within 1% error. This measurement confirms the expected preferential coherent excitation of the TM01 mode over the HEM11 mode, which lies nearby in frequency but still decisively outside the error estimate established over multiple measurements. The predicted frequency shift of 100 GHz under a change in DLW wall thickness of 50 µm is also observed, as shown in Fig. 4.8. The measurement demonstrates the modular tunability of the source; a discretely tunable radiation source can be created by assembling a “magazine” of different tubes and exchanging them in situ to switch between frequencies. The small size and relative ease of manufacture of the DLW used here is particularly conducive to this method.

109

Table 4.2 Summary of measured quantities and their comparison with analytical and oopic pro results.

b = 350 µm measured

analytical

oopic

f (GHz)

367 ± 3

368

365

U (µJ)

10 ± 2

10.3

12

—

58

—

f (GHz)

268 ± 2

273

266

U (µJ)

10 ± 2

15

18

> 60

84

—

tpulse (ps) b = 400 µm

tpulse (ps)

The observed level of radiated energy agrees reasonably with predictions as well. This serves as verification that the CCR process is able to extract an appreciable amount of energy from the driving beam in this configuration, making it a useful option to consider for implementation as a THz source in a user facility or in support of a particular experiment. It is interesting to consider structure variations that, while not compatible with Neptune beam parameters, would give improved results when paired with other existing beams. For example, with a beam as short as σz ∼ 20 µm, a tube made of HDPE with a wall thickness of 30 µm and inner radius of 50 µm would radiate at a frequency of f ≈ 1.8 THz. HDPE is chosen for this example because the absorption coefficient of fused silica is prohibitively high at frequencies above 1 THz (Section 3.2.1). The total radiated energy would be an estimated 30 mJ for a beam charge of 0.5 nC and a 10 cm-long dielectric structure, accounting for absorption losses, corresponding to a peak power on the order of 50 MW in a pulse 450 ps long. An advantage

110

Peak Power Comparison

108

1 .8 THz peak power ( W )

107

106

105

104

1 - 3 THz

JLab

NovoFEL

0.27 THz 0.37 THz 1 - 5 THz UCSB

Figure 4.9

broad band

Neptune

hypothetical

Comparison of peak power achievable at THz source facilities.

of an increased structure length is a longer pulse duration and thus a lower bandwidth. For this hypothetical case, ∆f is ∼ 0.1%. The same peak power would be obtained using a shorter structure, albeit with a larger bandwidth. The electron beam energy need only be sufficient to allow for deceleration and for focusing of the beam through the tube, nominally on the order of a few hundred MeV to 1 GeV if a typical transverse emittance is assumed. The use of DLW structures at high power is limited by an electric breakdown threshold, which e.g. for fused silica tubing has been found experimentally to be ∼13.8 GV/m (at the inner dielectric surface) [88]. The case study presented here is within this limit, neglecting to consider sub-breakdown material degradation effects and possible lowering of breakdown threshold due to long-pulse exposure. Fig. 4.9 compares the performance of the Neptune and hypothetical CCR THz sources with existing state-of-the-art FEL and synchrotron THz light source facilities. The comparison is drawn in terms of peak power, as the scaling of the DLW CCR source to high repetition rates for high average power has not been studied; it is assumed that pulsed heating effects would play a limiting role in the operation of a DLW-based source at a high rep-rate. We see that the peak narrow-

111

Table 4.3 Parameters used in hypothetical case study.

beam charge

Q

0.5 nC

RMS bunch length

σz

20 µm

RMS beam size

σr

10 µm

tube length

L

10 cm

inner radius

a

50 µm

outer radius

b

80 µm

dielectric constant (HDPE)

εr

2.25

band THz power achievable in a CCR scheme implemented in the hypothetical case would be considerably higher than that produced by the Novosibirsk THz FEL [135, 136] and the Jefferson Lab broadband CSR THz source [60]. Highpower, beam-synchronized THz radiation at a specific frequency is a tool that has enabled intriguing experimental research. One example is a pump-probe variety of experiments utilizing an FEL X-ray beam to probe a sample that has been pumped into an excited state with a THz pulse, such as that reported by Cavalleri et al. [137]. Conversely, the work of Huber et al. [138] uses a THz pulse as a probe. Work by Cole et al. [139] demonstrates the manipulation of quantum bits using narrow-band THz radiation produced at the UCSB THz FEL [140]. Flexible narrow-band sources are also of interest for THz FEL/IFEL seeding [141]. This study represents a successful adaptation of the previously proven Cherenkov FEL concept to the realm of intense, ultra-short electron beams. While the radiated energy and frequency achievable in the measurements presented here were limited by available beam parameters, the results prove the potential of the

112

method to produce tunable, narrow-band, pulse-length variable, multi-megawatt peak-power radiation at f > 1 THz in existing modern electron accelerators.

113

APPENDIX

A

Coherent Edge Radiation from a Compact Bend Magnet

The study of radiation from charged particles in the uniform magnetic field of a dipole bending magnet is a primary technical consideration in the design of circular accelerator machines, and was first observed in this context, leading to use of the term synchrotron radiation (SR). A complementary form of radiation, emitted at the edge of a bending magnet as a beam enters, is termed edge radiation (ER). These forms of radiation are invariably found together because of the nature of their origin.

A.1

Synchrotron and Edge Radiation

The difference between synchrotron and edge radiation is difficult to define in a general way. Edge radiation is named so because it is studied in the specific context of charged particles passing from a region of zero magnetic field into the uniform field of a dipole bending magnet, and is treated mathematically as a particle traveling from a force-free region, past a short spatially-varying field region, and into a uniform field [142], as illustrated in Fig. A.1; it can therefore be thought of as the superposition of radiation from a constant-velocity particle

114

free space

edge

n

e

A

θx

x y

dipole magnet

P

β

B

z

B0 0 no field

Figure A.1

fringe

uniform field

A charged particle passing from a zero-field region into a dipole

magnet.

x y

n θx

z

Figure A.2

e

P

β

An electron in uniform circular motion.

under sudden longitudinal acceleration and a transient instance of synchrotron radiation, which by definition arises from particle motion in a uniform, constant magnetic field. We will discuss the qualitative differences between SR and ER that warrant their distinction. The instantaneous SR angular intensity pattern, as seen by a stationary observer located at point P in Fig. A.2, is azimuthally symmetric [143] about the electron velocity direction β. Fig A.3 shows the dependence of the intensity, integrated over all frequencies, on observation angle for polarization components both parallel to the plane of motion (σ component) and perpendicular to it (π component) [144]. It is seen that the radiation is predominantly linearly polarized in the plane of motion and peaked in the β direction. In addition, the intensity is concentrated within an angle θ = 1/γ from the instantaneous direction of motion.

115

intensity ( a.u. )

σ polarization π polarization

-3

Figure A.3

-2

-1

0 γθ

1

2

3

Synchrotron radiation angular intensity patterns for in-plane (σ) and

out-of-plane (π) polarization components.

In contrast, the edge radiation angular intensity distribution, as modeled by Bosch [132, 145] and Chubar [146], is a hollow pattern peaked at an angle θ = 1/γ from the β direction. The radiation is largely radially polarized, as seen from the comparable relative magnitudes of calculated intensity patterns of the orthogonal polarization components shown in Fig. A.4. These characteristics, which are reminiscent of transition radiation or “soft bremsstrahlung,” [147] arise from the strongly longitudinal nature of the interaction; although the magnetic force is perpendicular to the particle motion at every point, the relativistic particle transitions very quickly from its initially straight trajectory to the final state of transverse motion, effectively experiencing a change in the longitudinal component of its velocity. The patterns in Fig. A.4 are computed using a model in which the edge region is taken to be a region of smoothly-varying magnetic field with a finite length [148]. In the long-wavelength limit, for which the radiation formation length λγ 2 exceeds the characteristic length of the edge region, the ER is similar in nature to transition radiation. In practice, the angular distribution of SR is is heavily weighted to one side

116

γ -1

γ -1 -γ -1 θ x

θy

σ polarization Figure A.4

-γ -1 θx

θy

π polarization

θy

θx

σ+π

Edge radiation angular intensity patterns, using finite-edge-length

model in the long-wavelength limit.

due to the integrated effect of the sweeping of the “searchlight” radiation cone over the observation window. The ER intensity, however, is emitted only from the point where the charge enters the magnet edge and thus is confined to a narrow cone. This fact can allow the ER signal to be distinguished experimentally, with a minimum of SR pollution, by using a detector placed facing opposite β with an angular field of view limited to ∼ 2/γ. The frequency spectrum of SR or ER from a single particle is broadband, extending to an upper critical frequency above which the radiated intensity is negligible. The SR critical frequency is given by [15]   3 3 c ωc = γ , 2 ρ

(A.1)

where ρ is the instantaneous radius of curvature of the particle trajectory. As is apparent from the γ 3 factor, a high-energy particle will produce a very broad range of frequencies. Comparing the single-particle spectra for the SR and ER (finite-edge-length model) cases [148], it is seen that the ER spectrum is roughly flat up to the critical frequency relative to the SR spectrum. This approximation enables the electron bunch length measurement presented in the following section.

117

window dipole magnet

compressed electron beam

radiation

ICT

Figure A.5

(a)

OAP mirror

Diagram of experimental setup for CER measurements.

(b)

polarization measurement setup

interferometric setup Golay cell

Golay cell Michelson interferometer polarizer window window

Figure A.6

A.2

Diagram of CER diagnostic setup.

CER Measurements at Neptune

The source of edge radiation studied here is unique in that it contains only one magnet edge. Previous measurements, usually taken in a chicane or other system containing multiple magnets, are affected by the interference of CER from multiple magnet edges [149, 150, 145]. In addition, because of the location of the detector and the strong bend angle, this setup suffers relatively little from pollution by CSR content in the measurements when properly aligned. Therefore, this system presents an opportunity to study edge radiation in a more straightforward manner than has been possible in the past. The setup used in the measurements, pictured in Fig. A.5, is a modification

118

CER energy vs. bunch charge

signal (mV)

150

data quadratic fit

100

50

0 50

Figure A.7

100

150 200 charge (pC)

250

300

Plot of measured CER energy as a function of electron bunch charge.

of that used in the previously described Cherenkov radiation experiment (Fig. 3.7). The incoming compressed electron beam enters the dipole magnet and is bent strongly upward by an angle of 90 degrees, in a bending radius ∼ 3 cm. Emitted radiation continues forward to an OAP mirror, where it is collimated and directed out of the vacuum box through the Z-cut crystal quartz window. Diagnostics for measuring the polarization and autocorrelation of the radiation pulse are diagrammed in Fig. A.6. An ICT placed upstream of the magnet measures the bunch charge. Fig. A.7 shows the measured CER energy as a function of beam charge, which exhibits the expected quadratic dependence. The primary goal of these measurements was a determination of the electron bunch length σz . First, it was necessary to verify that the observed radiation was indeed dominated by CER, as outlined below. The autocorrelation of the radiation pulse was then measured and used to infer the bunch length.

119

Polarization Scan

signal ( mV )

50 40

data

30

fit

20

data, steering corrected fit

10 0 0

50

100

150

polarizer angle ( deg. )

Figure A.8

Measured CER signal vs. polarizer angle, for two incident-beam

steering configurations.

A.2.1

Polarization

Polarization measurements were used as a tool to distinguish CER from CSR. By rotating the wire-grid polarizer through an angle of 180 degrees and comparing the relative intensities of the σ- and π-polarized components, we were able to determine qualitatively the relative strength of CSR present in the signal. A large sinusoidal dependence of the detected signal on the polarizer angle was taken as a sign of CSR-dominated radiation. Since the vacuum window aperture limits the total angular field of view to . 2/γ, a strong CSR presence indicates that the beam trajectory vector coming into the dipole magnet is not collinear with the observation direction defined by the optical transport line, the window, and the detector. In this case, the electron beam steering was adjusted slightly and the polarization scan repeated; after a few iterations, an approximately flat polarization was achieved. Fig. A.8 compares a polarizer scan from an initially un-corrected beam steering state to that measured after steering correction.

120

A.2.2

Compressed Electron Bunch Length

Longitudinal compression of the electron beam was central to the success of the CCR experiment, because appreciable coherent excitation at the design frequency required an RMS bunch length several times shorter than that of the nominal uncompressed bunch (∼ 3 ps). Therefore it was necessary to measure the bunch length at the interaction point in order to verify its usability and eliminate it as a potential source of difficulty. The method used to unfold the bunch length information from the measured autocorrelation interferogram is identical to that applied by Murokh et al. [32] to coherent transition radiation measurements. The technique uses a “tri-Gaussian” functional form, parameterized in terms of the bunch duration σt and a characteristic frequency cutoff ξ = ωc−1 (imposed by the measurement apparatus): s(τ ) ∝ e−τ

2 /4σ 2 t

−p

2σt σt2

+ ξ2

e−τ

2 /4(σ 2 +ξ 2 ) t

σt 2 2 2 +p e−τ /4(σt +2ξ ) . 2 σt + 2ξ 2

(A.2)

By fitting this form to the measured autocorrelation trace in the time domain, taking ξ as an adjustable fit parameter, a σt value is obtained. The applicability of this fitting function assumes the flat frequency spectrum typical of CTR. Relying on the approximation that the CER spectrum is flat in the wavelength range of interest, we can use the same technique to obtain the electron bunch length from our CER autocorrelation measurements. A measured CER autocorrelation trace is shown in Fig. A.9a. The data is apodized [133] and fit to Eq. A.2, as shown in Fig. A.9b. The temporal bunch duration is thus determined to be σt = 550 ± 50 fs, which corresponds to a bunch length of σz = 165 ± 15 µm. This value is used in discussion and predictions throughout this dissertation.

121

CER Autocorrelation

( a.u. )

(a)

raw data apodization -10

-5

0 delay ( ps )

5

10

CER Autocorrelation

(b)

apodized data tri-Gaussian

( a.u. )

σt ≈ 550 ± 50 fs

-10

Figure A.9

-5

0 delay ( ps )

5

10

(a) Raw measured autocorrelation interferogram of CER pulse, with

Gaussian apodization function. (b) Apodized interferogram with tri-Gaussian fit for determining σt .

122

APPENDIX

B

Electromagnetic Wave Equation

The vector electromagnetic fields E and H in regions of space containing no free charge or current are governed by the source-free Maxwell’s Equations ∇·E=0

(B.1)

∇·H=0

(B.2)

∇ × E = −µ ∇×H=ε

∂H ∂t

∂E ∂t

(B.3) (B.4)

where µ and ε represent the magnetic permeability and electric permittivity respectively. We may derive from these the wave equation describing the propagation of EM radiation in the following manner. We begin with Eq. B.3, taking the curl of both sides to obtain ∇ × (∇ × E) = −µ

∂ (∇ × H). ∂t

(B.5)

Making use of the vector identity ∇ × (∇ × V) = ∇(∇ · V) − ∇2 V

123

(B.6)

and Eq. B.4, we can rewrite both sides of Eq. B.5 to get ∂2E :0  ∇( ∇·  E) − ∇2 E = −µε 2 ∂t   1 ∂2 2 −→ ∇ − 2 2 E = 0. c ∂t

(B.7)

A wave equation for the magnetic field H can be found in a similar way. It many instances it is useful to have a wave equation for a single field component. Suppose that E is written in Cartesian coordinates as E = Ex x ˆ +Ey y ˆ +Ez ˆ z. Inserting this into Eq. B.7 and making use of D’Alembertian operator notation   1 ∂2 2  ≡ ∇ − c2 ∂t2 , we have Ex x ˆ + Ey y ˆ + Ez ˆ z = 0.

(B.8)

Because the three vector terms on the LHS of this expression are orthogonal, it is clear that each must be zero independently in order to satisfy the equation. Therefore we can separate them into three wave equations, one for each of the three field components. It is important to note that each component is still a scalar function of all three spatial dimensions and time, i.e. Ez (x, y, z, t). In this dissertation we are concerned with guided waves in a translationally invariant system, and so it will be convenient to separate out the axial and time dependences. This is accomplished by using a standard separation of variables approach as follows. We begin by assuming that the field component Ez can be written as a product of three functions Ez = Ez,⊥ (x, y)Z(z)T (t). The wave equation for Ez is then   1 ∂2 2 0 = ∇ − 2 2 Ez,⊥ (x, y)Z(z)T (t) c ∂t = ZT ∇2⊥ Ez,⊥ + Ez,⊥ T ∇2z Z −

124

Ez,⊥ Z d2 T , c2 dt2

(B.9)

where the Laplacian has been separated into transverse and longitudinal components as ∇2 = ∇2⊥ + ∇2z . Dividing through by Ez,⊥ ZT and grouping terms, we get 

   1 1 2 1 d2 T 2 = 0. ∇ Ez,⊥ + ∇z Z − 2 Ez,⊥ ⊥ Z c T dt2

(B.10)

Recognizing that each of the two grouped terms on the LHS of Eq. B.10 are independent of each other and therefore must be equal to the same constant for the equation to be true, we introduce the arbitrary separation constant −k 2 and write 

   1 1 2 1 d2 T 2 ∇ Ez,⊥ + ∇z Z = 2 = −k 2 . Ez,⊥ ⊥ Z c T dt2

(B.11)

This can be separated into two equations 1 1 ∇2⊥ Ez,⊥ + ∇2z Z = −k 2 Ez,⊥ Z d2 T = −c2 k 2 T, dt2

(B.12)

(B.13)

the latter of which is easily recognized as the differential equation for sinusoidal functions. The solution thus describes the time dependence of the fields, T (t) = eickt = eiωt .

(B.14)

We rearrange Eq. B.12 by again grouping independently varying terms and introducing another separation constant kz2 :   1 −1 d2 Z 2 2 ∇⊥ Ez,⊥ + k = = kz2 . Ez,⊥ Z dz 2

(B.15)

This yields the two equations   2 ∇⊥ + (k 2 − kz2 ) Ez,⊥ = 0 d2 Z = −kz2 Z, 2 dz

125

(B.16) (B.17)

the last of which gives the standard longitudinal dependence Z(z) = eikz z .

(B.18)

Eq. B.16 is the wave equation describing the transverse variation of the longitudinal field Ez . The quantity (k 2 − kz2 ) represents the transverse component of the wavenumber.

126

References

[1] P. A. Cherenkov, “Visible Radiation Produced by Electrons Moving in a Medium with Velocities Exceeding that of Light,” Phys. Rev. 52, 378 (1937). [2] A. Chao, Handbook of Accelerator Physics and Engineering, World Scientific (1999). [3] S. Y. Lee, Accelerator Physics, World Scientific 2nd edition (2004). [4] M. Reiser, Theory and Design of Charged Particle Beams, 2nd Ed., Wiley (2008). [5] J. B. Rosenzweig, Fundamentals of Beam Physics, Oxford University Press (2003). [6] T. P. Wangler, RF Linear Accelerators, Wiley-VCH 2nd edition (2008). [7] H. Wiedemann, Particle Accelerator Physics, 3rd Ed., Springer (2007). [8] P. Musumeci et al., “Experimental Generation and Characterization of Uniformly Filled Ellipsoidal Electron-Beam Distributions,” Phys. Rev. Lett. 100, 244801 (2008). [9] J. B. Rosenzweig et al., “Emittance Compensation with Dynamically Optimized Photoelectron Beam Profiles,” Nucl. Inst. and Meth. A 557, 87–93 (2006). [10] Y. Li and J. Lewellen, “Generating a Quasiellipsoidal Electron Beam by 3D Laser-Pulse Shaping,” Phys. Rev. Lett. 100, 074801 (2008). [11] V. Petrillo, L. Serafini, and P. Tomassini, “Ultrahigh Brightness Electron Beams by Plasma-Based Injectors for Driving All-Optical Free-Electron Lasers,” Phys. Rev. ST Accel. Beams 11, 070703 (2008).

127

[12] R. Ganter et al., “Laser-Photofield Emission from Needle Cathodes for Low-Emittance Electron Beams,” Phys. Rev. Lett. 100, 064801 (2008). [13] S. G. Anderson, Creation, Manipulation, and Diagnosis of Intense, Relativistic Picosecond Photo-electron Beams, Ph.D. thesis, University of California, Los Angeles (2002). [14] R. Y. Tsien, “Pictures of Dynamic Electric Fields,” Am. J. Phys. 40, 46–56 (1972). [15] J. D. Jackson, Classical Electrodynamics, 3rd Ed., Wiley (1999). [16] C. J. Hirschmugl et al., “Multiparticle Coherence Calculations for Synchrotron-Radiation Emission,” Phys. Rev. A 44, 1316–1320 (1991). [17] J. Nodvick and D. Saxon, “Suppression of Coherent Radiation by Electrons in a Synchrotron,” Phys. Rev. 96, 180 (1954). [18] G. Collins and V. Reiling, “Cerenkov Radiation,” Phys. Rev. 54, 499–505 (1938). [19] H. Wyckoff and J. Henderson, “The Spatial Asymmetry of Cerenkov Radiation as a Function of Electron Energy,” Phys. Rev. 64, 1–7 (1943). [20] B. Bolotovskii, Sov. Phys.-Usp. 4, 781 (1962). [21] J. Neighbours, F. Buskirk, and A. Saglam, “Cherenkov Radiation from a Finite-Length Path in a Gas,” Phys. Rev. A 29, 3246–3252 (1984). [22] T. Ypsilantis and J. Seguinot, “Theory of Ring Imaging Cherenkov Counters,” Nucl. Inst. and Meth. A 343, 30–51 (1994). [23] J. V. Jelley, Cherenkov Radiation and Its Applications, Pergamon Press (1958). [24] B. Baker and E. Copson, The Mathematical Theory of Huygens’ Principle, Oxford University Press (1969). [25] I. Frank and I. Tamm, Dokl. Akad. Nauk 14, 109 (1937). [26] I. Tamm, Zh. fis. 1, 439 (1939). [27] M. Abele, Nuovo Cim. 9, Suppl. (3) 207 (1952). [28] I. Frank and V. Ginzburg, J. Phys. USSR 9, 353 (1945).

128

[29] T. Takahashi et al., “Coherent Transition Radiation at Submillimeter and Millimeter Wavelengths,” Phys. Rev. E 48, 4674–4677 (1993). [30] L. Wartski et al., “Interference Phenomenon in Optical Transition Radiation and its Application to Particle Beam Diagnostics and MultipleScattering Measurements,” J. Appl. Phys. 46, 3644–3653 (1975). [31] U. Happek, A. J. Sievers, and E. B. Blum, “Observation of Coherent Transition Radiation,” Phys. Rev. Lett. 67, 2962–2965 (1991). [32] A. Murokh et al., “Bunch length measurement of picosecond electron beams from a photoinjector using coherent transition radiation,” Nucl. Inst. and Meth. A 410, 452–460 (1998). [33] Y. Shibata et al., “Diagnostics of an Electron Beam of a Linear Accelerator Using Coherent Transition Radiation,” Phys. Rev. E 50, 1479–1484 (1994). [34] Y. Shibata et al., “Coherent Transition Radiation in the Far-Infrared Region,” Phys. Rev. E 49, 785–793 (1994). [35] J. B. Rosenzweig, G. Travish, and A. Tremaine, “Coherent Transition Radiation Diagnosis of Electron Beam Microbunching,” Nucl. Inst. and Meth. A 365, 255–259 (1995). [36] A. Tremaine et al., “Observation of Self-Amplified Spontaneous-EmissionInduced Electron-Beam Microbunching Using Coherent Transition Radiation,” Phys. Rev. Lett. 81, 5816–5819 (1998). [37] S. Schelkunoff, “On Diffraction and Radiation of Electromagnetic Waves,” Phys. Rev. 56, 308–316 (1939). [38] J. Stratton and L. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99–107 (1939). [39] Y. Shibata et al., “Observation of Coherent Diffraction Radiation from Bunched Electrons Passing Through a Circular Aperture in the Millimeterand Submillimeter-Wavelength Regions,” Phys. Rev. E 52, 6787–6794 (1995). [40] D. Xiang and W. Huang, “Properties of Diffraction Radiation in Practical Conditions: Finite Size Target Effect, Surface Roughness and Pre-Wave Zone,” Nucl. Inst. and Meth. B 248, 163–169 (2006). [41] A. Potylitsyn, “Thomson Scattering of Coherent Diffraction Radiation by an Electron Bunch,” Phys. Rev. E 60, 2272–2279 (1999).

129

[42] R. B. Fiorito and D. W. Rule, “Diffraction radiation diagnostics for moderate to high energy charged particle beams,” Nucl. Inst. and Meth. B 173, 67–82 (2001). [43] D. Xiang and W. Huang, “Ultrashort Electron Bunch Length Measurement with Diffraction Radiation Deflector,” Phys. Rev. ST Accel. Beams 10, 012801 (2007). [44] A. Potylitsyn et al., “Coherent Radiation Recoil Effect for the Optical Diffraction Radiation Beam Size Monitor at SLAC FFTB,” Nucl. Inst. and Meth. B 227, 170–174 (2005). [45] K. L. F. Bane and G. Stupakov, “Transition Radiation Wakefields for a Beam Passsing Through a Metallic Foil,” Phys. Rev. ST Accel. Beams 7, 064401 (2004). [46] W. Gai et al., “Experimental Demonstration of Wake-Field Effects in Dielectric Structures,” Phys. Rev. Lett. 61, 2756–2758 (1988). [47] J. B. Rosenzweig et al., “Experimental Observation of Plasma Wake-Field Acceleration,” Phys. Rev. Lett. 61, 98 (1988). [48] P. B. Wilson, “Wakefield Accelerators,” SLAC Summer Institute on Particle Physics (1985). [49] J. T. Seeman, “Collective Electron Driven Linac for High Energy Physics,” Trans. Nuclear Sci. 30, 3180–3182 (1983). [50] K. L. F. Bane et al., “On Collinear Wakefield Acceleration,” Trans. Nuclear Sci. 32, 3524–3526 (1985). [51] R. J. England, “Generation and Measurement of Relativistic Electron Bunches Characterized by a Linear Ramped Current Profile,” Phys. Rev. Lett. 100, 214802 (2008). [52] J. G. Power et al., “Wakefield Excitation in Multimode Structures by a Train of Electron Bunches,” Phys. Rev. E 60, 6061–6067 (1999). [53] G. P. Williams, “Filling the THz Gap – High Power Sources and Applications,” Rep. Prog. Phys. 69, 301–326 (2006). [54] A. Redo-Sanchez and X. Zhang, “Terahertz Science and Technology Trends,” IEEE Selected Topics in Quantum Electronics 14, 260–269 (2008).

130

[55] R. Kohler, “Terahertz Semiconductor-Heterostructure Laser,” Nature 417, 156–159 (2002). [56] B. S. Williams, “Terahertz Quantum-Cascade Lasers,” Nature Photonics 1, 517–525 (2007). [57] D. Auston, “Picosecond Optoelectronic Switching and Gating in Silicon,” Appl. Phys. Lett. 26, 101–103 (1975). [58] S. Tochitsky et al., “Generation of Megawatt-Power Terahertz Pulses by Noncollinear Difference-Frequency Mixing in GaAs,” J. Appl. Phys. 98, 026101 (2005). [59] D. P. Akitt, W. Q. Jeffers, and P. D. Coleman, “Water Vapor Gas Laser Operating at 118-Microns Wavelength,” Proc. IEEE 54, 547–551 (1966). [60] G. L. Carr et al., “High-Power Terahertz Radiation from Relativistic Electrons,” Nature 420, 153 (2002). [61] J. M. Byrd et al., “Observation of Broadband Self-Amplified Spontaneous Coherent Terahertz Synchrotron Radiation in a Storage Ring,” Phys. Rev. Lett. 89, 224801 (2002). [62] E.P. Garate et al., “Cherenkov free electron laser operation from 375 to 1000 micron,” Nucl. Inst. and Meth. A 259(1-2), 125 – 127 (1987). [63] S. J. Smith and E. M. Purcell, “Visible Light from Localized Surface Charges Moving across a Grating,” Phys. Rev. 92(4), 1069 (1953). [64] S. E. Korbly et al., “Observation of Frequency-Locked Coherent Terahertz Smith-Purcell Radiation,” Phys. Rev. Lett. 94, 054803 (2005). [65] B. B. Hu and M. C. Nuss, “Imaging with Terahertz Waves,” Opt. Lett. 20, 1716–1718 (1995). [66] D. M. Mittleman et al., “Recent Advances in Terahertz Imaging,” Appl. Phys. B 68, 1085–1094 (1999). [67] M. Tonouchi, “Cutting-Edge Terahertz Technology,” Nature Photonics 1, 97–105 (2007). [68] K. Sakai and M. Tani, “Introduction to Terahertz Pulses,” Terahertz Optoelectronics, Topics Appl. Phys. 97, 1–30 (2005). [69] B. Ferguson and X. Zhang, “Materials for Terahertz Science and Technology,” Nature Materials 1, 26–33 (2002).

131

[70] N. Laman et al., “High-Resolution Waveguide THz Spectroscopy of Biological Molecules,” Biophys J. 94, 1010–1020 (2008). [71] R. Rungsawang et al., “Angle-Dependent Terahertz Time-Domain Spectroscopy of Amino Acid Single Cyrstals,” J. Phys. Chem. B 110, 21259 (2006). [72] N. Brandt et al., “Low-Frequency Vibrational Motions in Proteins: Physical Mechanisms and Effects on Functioning,” Eur. Phys. J. B 65, 419–424 (2008). [73] B. Fischer, M. Walther, and P. Jepsen, “Far-Infrared Vibrational Modes of DNA Components Studied by Terahertz Time-Domain Spectroscopy,” Phys. Med. Biol. 47, 3807–3814 (2002). [74] C. T. M. Chang and J. W. Dawson, “Propagation of Electromagnetic Waves in a Partially Dielectric Filled Circular Waveguide,” J. Appl. Phys. 41, 4493–4500 (1970). [75] J. Walsh et al., “Preliminary Results from a Microwave-Driven 100 um Wavelength Cherenkov FEL Experiment,” Nucl. Inst. and Meth. A 272, 132–135 (1988). [76] P. Chen, “Acceleration of Electrons by the Interaction of a Bunched Electron Beam with a Plasma,” Phys. Rev. Lett. 54, 693–696 (1985). [77] I. Blumenfeld et al., “Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator,” Nature 445, 741 (2007). [78] M. Hogan et al., “Multi-GeV Energy Gain in a Plasma-Wakefield Accelerator,” Phys. Rev. Lett. 95, 054802 (2005). [79] C. Joshi, “Plasma Accelerators,” Sci. Am. 294, 41–47 (2006). [80] C. Joshi, “The Development of Laser- and Beam-Driven Plasma Accelerators as an Experimental Field,” Phys. Plasmas 14, 055501 (2007). [81] S. Tochitsky et al., “Enhanced Acceleration of Injected Electrons in a Laser-Beat-Wave-Induced Plasma Channel,” Phys. Rev. Lett. 92, 095004 (2004). [82] M. Everett et al., “Trapped Electron Acceleration by a Laser-Driven Relativistic Plasma Wave,” Nature 368, 527–529 (1994).

132

[83] S. Mangles et al., “Laser-Wakefield Acceleration of Monoenergetic Electron Beams in the First Plasma-Wave Period,” Phys. Rev. Lett. 96, 215001 (2006). [84] F. Tsung et al., “Near-GeV-Energy Laser-Wakefield Acceleration of SelfInjected Electrons in a Centimeter-Scale Plasma Channel,” Phys. Rev. Lett. 93, 185002 (2004). [85] J. B. Rosenzweig et al., “Towards a Plasma Wake-Field Acceleration-Based Linear Collider,” Nucl. Inst. and Meth. A 410, 532–543 (1998). [86] J. G. Power et al., “Measurements of the longitudinal wakefields in a multimode, dielectric wakefield accelerator driven by a train of electron bunches,” Phys. Rev. ST Accel. Beams 3, 101302 (2000). [87] R J. England, Longitudinal Shaping of Relativistic Bunches of Electrons Generated by an RF Photoinjector, Ph.D. thesis, University of California, Los Angeles (2007). [88] M. C. Thompson et al., “Breakdown Limits on Gigavolt-per-Meter Electron-Beam-Driven Wakefields in Dielectric Structures,” Phys. Rev. Lett. 100, 214801 (2008). [89] R. Bonifacio, C. Pellegrini, and L. Narducci, “Collective Instabilities and High-Gain Regime in a Free-Electron Laser,” Opt. Commun. 50, 373–378 (1984). [90] A. Gover et al., “Time and Frequency Domain Analysis of Superradiant Coherent Synchrotron Radiation in a Waveguide Free-Electron Laser,” Phys. Rev. Lett. 72, 1192–1195 (1994). [91] A. Gover et al., “Superradiant and Stimulated-Superradiant Emission in Prebunched Electron-Beam Radiators. I. Formulation,” Phys. Rev. ST Accel. Beams 8, 030701 (2005). [92] A. Gover et al., “Superradiant and Stimulated-Superradiant Emission in Prebunched Electron-Beam Radiators. II. Radiation Enhancement Schemes,” Phys. Rev. ST Accel. Beams 8, 030702 (2005). [93] K. Ng, “Wake fields in a dielectric-lined waveguide,” Phys. Rev. D 42, 1819 (1990). [94] M. Rosing and W. Gai, “Longitudinal- and Transverse-Wake-Field Effects in Dielectric Structures,” Phys. Rev. D 42, 1829–1834 (1990).

133

[95] S. Y. Park and J. L. Hirshfield, “Theory of Wakefields in a Dielectric-Lined Waveguide,” Phys. Rev. E 62, 1266–1283 (2000). [96] C. Wang and J. L. Hirshfield, “Theory for Wakefields in a Multizone Dielectric Lined Waveguide,” Phys. Rev. ST Accel. Beams 9, 031301 (2006). [97] M. Ibanescu et al., “Analysis of Mode Structure in Hollow Dielectric Waveguide Fibers,” Phys. Rev. E 67, 046608 (2003). [98] P. Yeh et al., “Electromagnetic Propagation in Periodic Stratified Media. I. General Theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [99] P. Yeh et al., “Theory of Bragg Fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978). [100] D. Marcuse, Theory of Dielectric Optical Waveguides, Academic Press (1974). [101] J. B. Rosenzweig et al., “An Ultra-High Gradient Cerenkov Wakefield Acceleration Experiment at SLAC FFTB,” In Proceedings of Advanced Accelerator Concepts: Eleventh Workshop pp. 811 – 817 (2004). [102] M. Abramowitz and I. Stegun, editors, Handbook of Mathematical Functions, Dover (1970). [103] A. Tremaine and J. B. Rosenzweig, “Electromagnetic wake fields and beam stability in slab-symmetric dielectric structures,” Phys. Rev. E 56, 7204 (1997). [104] D. L. Bruhwiler et al., “Particle-in-cell simulations of plasma accelerators and electron-neutral collisions,” Phys. Rev. ST Accel. Beams 4, 101302 (2001). [105] J. B. Rosenzweig et al., “The Neptune photoinjector,” Nucl. Inst. and Meth. A 410, 437–451 (1998). [106] D. T. Palmer, The Next Generation Photoinjector, Ph.D. thesis, Stanford University (1998). [107] S. G. Anderson, “Nonequilibrium Transverse Motion and Emittance Growth in Ultrarelativistic Space-Charge Dominated Beams,” Phys. Rev. ST Accel. Beams 3, 094201 (2000). [108] S. G. Anderson, “Space-Charge Effects in High Brightness Electron Beam Emittance Measurements,” Phys. Rev. ST Accel. Beams 5, 014201 (2002).

134

[109] L. Serafini and J. B. Rosenzweig, “Envelope analysis of intense relativistic quasilaminar beams in rf photoinjectors: A theory of emittance compensation,” Phys. Rev. E 55, 7565 (1997). [110] B. E. Carlsten, “New Photoelectric Injector Design for the Los Alamos National Laboratory XUV FEL Accelerator,” Nucl. Inst. and Meth. A 285, 313–319 (1989). [111] R. Paschotta, (2008).

Encyclopedia of Laser Physics and Technology,

Wiley

[112] T. Nakajyo et al., “Quantum Efficiencies of Mg Photocathode Under Illumination with 3rd and 4th Harmonics Nd:LiYF4 Laser Light in RF Gun,” Jpn. J. Appl. Phys. 42, 1470–1474 (2003). [113] A. Doyuran et al., “Investigation of X-Ray Harmonics of the Polarized Inverse Compton Scattering Experiment at UCLA,” In Proceedings of the 2005 Particle Accelerator Conference p. 2303 (2005). [114] S. G. Anderson et al., “Horizontal Phase-Space Distortions Arising from Magnetic Pulse Compression of an Intense, Relativistic Electron Beam,” Phys. Rev. Lett. 91, 074803 (2003). [115] C. Petichakis et al., “The effect of a gap between dielectric and waveguide on the dispersion relation of the Cherenkov device,” Vacuum 77, 533–538 (2005). [116] M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy of silicate glasses and the relationship to material properties,” J. Appl. Phys. 102, 043517 (2007). [117] K. Crandall and D. Rusthoi, “TRACE 3-D Documentation,” Technical Report LA-UR-97-886, Los Alamos National Laboratory (1997). [118] S. Ganichev and W. Prettl, Intense Terahertz Excitation of Semiconductors, Oxford University Press (2006). [119] T. Milligan, Modern Antenna Design, McGraw-Hill (1985). [120] R. E. Collin, Field Theory of Guided Waves, McGraw-Hill (1960). [121] F. Sporleder and H-G. Unger, Waveguide tapers, transitions, and couplers, Peter Peregrinus Ltd. (1988). [122] L. W. Couch II, Digital and Analog Communications Systems, 6th Ed., Prentice Hall (2001).

135

[123] C. Shannon, “Communication in the Presence of Noise,” Proc. IRE 37, 10 – 21 (1949). [124] Boston Electronics Corporation, “Predicting the performance of a photodetector,” product manual (1996). [125] J. R. Hickey et al., “Modified Optical System for the Golay Detector,” Rev. Sci. Instrum. 40, 732 (1969). [126] M. J. E. Golay, “The Theoretical and Practical Sensitivty of the Pneumatic Infra-Red Detector,” Rev. Sci. Instrum. 20, 816 (1949). [127] A. Rogalski, Infrared Detectors, CRC Press (2000). [128] C. Wheatstone, “An account of several new instruments and processes for determining the constants of a voltaic circuit,” Philos. Trans. R. Soc. Pt. II, 303–327 (1843). [129] W. T. Welford and R. Winston, High Collection Nonimaging Optics, Academic Press (1989). [130] Infrared Laboratories, “Silicon Bolometer Instruction Manual: UCLA System Notes,” product manual (2003). [131] S. Tochitsky, Private communication. [132] R. A. Bosch, “Long-wavelength radiation along a straight-section axis in an electron storage ring,” Nucl. Inst. and Meth. A 386, 525 – 530 (1997). [133] L. Frohlich, “Bunch Length Measurements Using a Martin-Puplett Interferometer at the VUV-FEL,” Master’s thesis, University of Hamburg (2005). [134] A. E. Siegman, Lasers, University Science Books (1986). [135] N. G. Gavrilov et al., “Status of the Novosibirsk High-Power Terahertz FEL,” Nucl. Inst. and Meth. A 575, 54–57 (2007). [136] G. Kulipanov et al., “Research Highlights from the Novosibirsk 400 W Average Power THz FEL,” Terahertz Science and Technology 1, 107–125 (2008). [137] A. Cavalleri et al., “Tracking the motion of charges in a terahertz light field by femtosecond X-ray diffraction,” Nature 442, 664 – 666 (2006).

136

[138] R. Huber et al., “How many-particle interactions develop after ultrafast excitation of an electron-hole plasma,” Nature 414, 286 – 289 (2001). [139] B. E. Cole et al., “Coherent Manipulation of Semiconductor Quantum Bits with Terahertz Radiation,” Nature 410, 60–63 (2001). [140] G. Ramian, “The New UCSB Free-Electron Lasers,” Nucl. Inst. and Meth. A 318, 225–229 (1992). [141] C. Sung et al., “Seeded Free-Electron and Inverse Free-Electron Laser Techniques for Radiation Amplification and Electron Microbunching in the Terahertz Range,” Phys. Rev. ST Accel. Beams 9, 120703 (2006). [142] R. A. Bosch et al., “Infrared Radiation from Bending Magnet Edges in an Electron Storage Ring,” Rev. Sci. Instrum. 67, 3346–3349 (1996). [143] A. Hofmann, The Physics of Synchrotron Radiation, Cambridge University Press (2004). [144] J. Schwinger, “On the Classical Radiation of Accelerated Electrons,” Phys. Rev. 75, 1912 (1949). [145] R. A. Bosch, “Edge Radiation in an Electron Storage Ring,” Nuo. Cim. 20, 483–493 (1998). [146] O. V. Chubar and N. V. Smolyakov, “Generation of Intensive LongWavelength Edge Radiation in High-Energy Electron Storage Rings,” In Proceedings of the 1993 Particle Accelerator Conference (1993). [147] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Elsevier Science (1975). [148] O. V. Chubar and N. V. Smolyakov, “VUV Range Edge Radiation in Electron Storage Rings,” J. Optics (Paris) 24, 117–121 (1993). [149] M. P. Dunning, Coherent Terahertz Radiation Emitted by a Sub-Picosecond Electron Beam in a Magnetic Chicane, Ph.D. thesis, University of California, Los Angeles (2009). [150] G. Andonian et al., “Observation of Coherent Terahertz Edge Radiation from Compressed Electron Beams,” Phys. Rev. ST Accel. Beams 12, 030701 (2009).

137