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Applications of Radio Frequency Resonant Cavities in Ultrafast Electron Microscopy BY JOHN HOGAN B.S., University of Illinois at Urbana-Champaign, 1996 M.S., University of Illinois at Chicago, 2009 THESIS Submitted as partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Chicago, 2016 Chicago, Illinois Defense Committee: W. Andreas Schroeder, Chair and Advisor Alan Nicholls, Research Resources Center Danilo Erricolo, Electrical and Computer Engineering Christoph Grein Robert Klie This thesis is dedicated to my parents, Tony and Therese Hogan. ii Acknowledgment I would like to thank my advisor, W. Andreas Schroeder, for all of his assistance in learning concepts, performing experiments, and also editing this thesis. I thank all of my other committee members, Danilo Erricolo, Christoph Grein, Robert Klie, and Alan Nicholls, for their help and support, with special thanks to Professor Erricolo for allowing me to constantly borrow his extremely expensive engineering equipment. I would like to thank Joel Berger and Ben Rickman, members of my research group, who provided invaluable help along the way. Joel was instrumental in helping me understand his research in the Ultrafast Electron Microscopy project, which lays the foundation for much of my work completed in this thesis. Joel and Ben both enabled me to tackle any problem involving a computer, and Ben developed a dielectric copper coating process that I was able to successfully use in one of my experiments. Finally, I want to thank Kevin Lynch and Richard Frueh, who built many of the components used in my experiments with quality and speed, and were always willing to go out of their way to answer any of my questions and provide any help I needed. iii Table of Contents 1 Introduction: UEM and RF Cavities.............................................. 1 2 Theory ............................................................................................... 6 2.1 2.1.1 Complex Notation for Time-Harmonic Fields ................................. 7 2.1.2 Cylindrical Waveguide Field Solutions ......................................... 10 2.1.3 Circular Cylindrical Cavity Field Solutions ................................... 15 2.1.4 Cavity Power Loss.......................................................................... 21 2.2 3 Electrodynamics of Resonant Cavities ................................................... 6 Power Supply ........................................................................................ 28 2.2.1 Transmission Lines and the Coaxial Cable TEM Mode ................ 30 2.2.2 General Impedance Matching ........................................................ 38 2.2.3 Inductive Coupling ......................................................................... 43 2.2.4 Impedance Matching of an Inductively Coupled Cavity ............... 50 Experimental Set-Up...................................................................... 57 3.1 Summary of Main Design Considerations ............................................ 59 3.2 Pre-existing Components ...................................................................... 64 3.2.1 Laser Oscillator and Frequency Doubling ..................................... 64 3.2.2 Laser Amplifier .............................................................................. 66 3.2.3 UEM Column Design ..................................................................... 67 3.3 Compression Resonant Cavity (CRC) .................................................. 68 3.3.1 Operation ........................................................................................ 68 3.3.2 Design ............................................................................................. 75 3.3.3 Construction ................................................................................... 80 3.3.4 Characterization.............................................................................. 91 3.4 Drift Tube and Deflection Plates......................................................... 100 3.5 Detection Resonant Cavity (DRC) ...................................................... 102 3.5.1 Operation ...................................................................................... 102 3.5.2 Double Pulse Operation and DRC Calibration ............................ 122 3.6 Synchronization and Power................................................................. 127 iv Table of Contents 4 5 Experimental Results ................................................................... 130 4.1 Dielectric Axial Hole Effects .............................................................. 130 4.2 End Cap Hole Effects .......................................................................... 132 4.3 DRC Signal ......................................................................................... 135 4.3.1 Variation of Double Pulse Separation (Calibration) .................... 135 4.3.2 Variation of Current ..................................................................... 139 4.3.3 Compression ................................................................................. 140 Conclusion ..................................................................................... 143 5.1 Summary ............................................................................................. 143 5.2 Future Prospects .................................................................................. 146 References ............................................................................................ 148 Appendix I Mathematica Code for DRC Single Pulse Analysis ...... 151 Appendix II Mathematica Code for DRC Double Pulse Analysis ... 168 Appendix III Mathematica Code for DRC Unequal DP Analysis .... 175 v List of Figures Figure 1-1 Resonant Cavities in UEM .................................................................. 2 Figure 2-1 RCCC Geometry ................................................................................. 6 Figure 2-2 TM010 Mode Pattern .......................................................................... 20 Figure 2-3 TM010 Mode Current Pattern ............................................................. 21 Figure 2-4 Transmission Line Concept .............................................................. 30 Figure 2-5 Coaxial Cable Geometry ................................................................... 34 Figure 2-6 Low Frequency Network................................................................... 39 Figure 2-7 Inductive Coupling ............................................................................ 45 Figure 3-1 System Components .......................................................................... 57 Figure 3-2 Axial Hole Effect on CRC Frequency .............................................. 74 Figure 3-3 Axial Hole Effect on CRC E-Field ................................................... 75 Figure 3-4 CRC Components.............................................................................. 81 Figure 3-5 Coupling Loop Detail ........................................................................ 82 Figure 3-6 Cavity/Coupling Loop Detail ............................................................ 83 Figure 3-7 End Cap Detail .................................................................................. 84 Figure 3-8 YAG Dielectric Insert and Coated Fused Silica ............................... 86 Figure 3-9 Coated Fused Silica Cavity in DRC .................................................. 86 Figure 3-10 YAG Insert with Coupling Loop .................................................... 88 Figure 3-11 Vacuum Pass-Through Assembly ................................................... 89 Figure 3-12 Completed CRC Assembly ............................................................. 90 Figure 3-13 CRC Impedance .............................................................................. 93 Figure 3-14 Effect of Loop Area on Impedance ................................................. 95 Figure 3-15 Effect of Loop Area on Reflection Coefficient............................... 96 Figure 3-16 QZERO Output for CRC................................................................. 97 Figure 3-17 Deflector Plates ............................................................................. 101 Figure 3-18 Drift Tube Assembly ..................................................................... 102 Figure 3-19 Normalized TM010 H-Field ............................................................ 111 Figure 3-20 Pulse Transform Variation ........................................................... 113 vi List of Figures Figure 3-21 Single Pulse (Time-Domain) ........................................................ 115 Figure 3-22 Radial Dependence of Vectors in the Mode Projection Integral .. 116 Figure 3-23 Axial Dependence of Vectors in the Mode Projection Integral .... 117 Figure 3-24 Mode Projection and Cavity Length ............................................. 118 Figure 3-25 Mode Projection Integral and Gun Voltage .................................. 119 Figure 3-26 Mode Projection and Transfer Function ....................................... 120 Figure 3-27 Mode Projection Multiplied by the Transfer Function ................. 121 Figure 3-28 Pulse Duration Dependence of EMF ............................................ 122 Figure 3-29 Double Pulse Systems (Time-Domain) ........................................ 123 Figure 3-30 Double Pulse Transforms .............................................................. 125 Figure 3-31 Axial Dependence of DP Transform ............................................. 125 Figure 3-32 DP Separation Dependence of EMF ............................................. 126 Figure 4-1 Variation of Resonant Frequency with Axial Hole Size ................. 132 Figure 4-2 Variation of Resonant Frequency with End Cap Hole Size............ 133 Figure 4-3 Effects of End Cap Hole Diverging Field ....................................... 134 Figure 4-4 Variation of DRC Signal with DP Separation ................................ 138 Figure 4-5 Variation of DRC Signal with Current ........................................... 140 Figure 4-6 Electron Pulse Compression ........................................................... 141 vii List of Symbols and Abbreviations AG Analytic Gaussian BBO β-Barium Borate CCD Charge Coupled Device CRC Compression Resonant Cavity DP Double Pulse DRC Detection Resonant Cavity DTEM Dynamic Transmission Electron Microscope/Microscopy EM Electromagnetic EMF Electromotive Force FFT Fast Fourier Transform HW1/eM Half-Width at 1/e Maximum LBO Lithium Triborate PLL Phase Locked Loop RCCC Right Circular Cylindrical Cavity RF Radio Frequency RMS Root Mean Square RTP Rubidium Titanyl Phosphate SMA SubMiniature Version A TE Transverse Electric TEM Transverse Electromagnetic TM Transverse Magnetic UED Ultrafast Electron Diffraction UEM Ultrafast Electron Microscopy UIC University of Illinois at Chicago UV Ultraviolet YAG Yttrium Aluminum Garnet Yb:KGW Ytterbium-Doped Potassium Gadolinium Tungstate viii Summary Radio frequency resonant cavities are becoming important components in the emerging field of Ultrafast Electron Microscopy, which aspires to bring Dynamic Transmission Electron Microscopy into the realm of sub-nanometer spatial resolution combined with sub-picosecond time resolution. To accomplish this goal, the time-varying electromagnetic fields of resonant cavities can be utilized for temporal focusing of laser-driven ultrafast electron pulses to be used for imaging. In this thesis I present the design and construction of a laser- synchronized dielectric-filled radio frequency resonant cavity used for temporal focusing. I also present the theoretical groundwork, design, and construction of a second resonant cavity for the purposes of phase synchronization and direct electron pulse duration measurement. ix 1 Introduction: UEM and RF Cavities Radio frequency (RF) resonant cavity oscillators have many applications in engineering and physics, and are currently becoming valuable tools in the specific field of Ultrafast Electron Microscopy (UEM) [1]. UEM involves operating an ordinary electron microscope with short pulses of electrons instead of a steady electron beam, allowing the possibility of taking “snapshots” of dynamic processes at the nanometer scale [2]. This document details the use of such cavities within the framework of a larger overall project that is ongoing in the UEM group at the University of Illinois at Chicago (UIC). This project uses a pulsed laser to drive a photoelectron gun, generating short electron pulses which are to be focused at a specimen (perturbed immediately before the arrival of an electron pulse by a separate, synchronized branch of the pulsed laser beam) for the purpose of developing a single-shot Dynamic Transmission Electron Microscope (DTEM). Such a research instrument will allow for a single pulse of electrons to create a clear image of irreversible dynamic processes at the nanometer scale, which is currently not possible [3-5]. The space-time resolution goal for the ultrafast DTEM is in the sub 1 nm-ps (nanometer-picosecond) range, with each pulse needing to contain roughly 108 electrons for proper imaging [68]. In this thesis, I detail the application of a cylindrical RF resonant cavity as an RF pulse compression cavity in the UIC UEM system. A simple schematic for a typical UEM system is shown in Figure 1-1. The electron pulse (shown in blue in Figure 1-1 at different stages of propagation) generated by photoemission in the laser-driven electron gun has intrinsic velocity dispersion (due to a variety of excess energies inherent to the photoemission process) and, possibly, spacecharge effects that will increase its duration during propagation to the specimen. The purpose of an RF compression cavity (labelled in Figure 1-1 as 1 the compression resonant cavity (CRC)) is to overcome this intrinsic pulse broadening in the longitudinal dimension by reversing the internal velocities of the electrons in the pulse, causing its duration to decrease (pulse compression) as it propagates, with the goal of providing the shortest pulse at the sample for the best space-time resolution in UEM. Currently, an RF resonant cavity has been successfully used for compression in Ultrafast Electron Diffraction (UED), which only requires up to 106 electrons per pulse instead of the 108 electrons required for imaging [9]. Figure 1-1 Resonant Cavities in UEM Schematic for a typical UEM system, with the addition of the DRC which is unique to the UIC system. Electronics are used to synchronize the CRC with the pulsed laser that is driving the photoelectron gun, to ensure that the electromagnetic (EM) field 2 inside the cavity is at the proper phase for compression when each pulse arrives. Depending on the phase of the field upon pulse arrival, the CRC could act to compress, stretch, accelerate or decelerate the pulse; thus proper synchronization is critical. The CRC described in this work is filled with low-loss dielectric material with an axial hole to allow for the passage of the pulsed electron beam. The dielectric material, in this case yttrium aluminum garnet (YAG), reduces the size of the cavity which makes it much easier to mount into the UEM column. The dielectric also allows for a polishing and copper coating process which can significantly improve performance. The use of dielectric-filled resonant cavities in single-shot UEM is unique to the UIC project. Power is delivered to the CRC by a 30 W RF amplifier, operating at one of the resonant frequencies of the RF cavity, and a low-loss coaxial cable with a small wire loop fixed to the end. The loop is inserted into the CRC and power from the amplifier drives a large current around the loop at the resonant frequency, which stimulates the corresponding resonant mode EM field inside the cavity. This is known as inductively coupling the cavity to the power source. The electric part of the stimulated EM field applies the necessary compression force to each electron pulse passing through the RF cavity. Proper impedance matching of the CRC and power source is critical to efficient power delivery. The design and application of a second cylindrical RF resonant cavity, labelled the Detection Resonant Cavity (DRC) in Figure 1-1, will also be detailed in this thesis. The DRC is not a standard component of any current UEM system other than the UIC system. I developed the concept of the DRC during an attempt to directly measure the duration of the compressed pulses at the temporal focal point. I developed theory which shows that a traveling electron pulse can be modelled as a current source which carries power at all frequencies, so that as a pulse travels through an RF resonant cavity (the DRC) it will stimulate multiple resonant modes of the cavity. In this case a coupling loop in the cavity can be 3 used as a receiving antenna, and magnetic flux from an impedance matched stimulated mode will create an EMF in the loop. This low power signal can be amplified and viewed on a fast oscilloscope. The theory shows that the amplitude of this signal can determine the time duration of the pulse passing through the DRC, without disturbing the pulse in any way. Unfortunately, the employed GHz-level DRC can only detect pulse durations for pulses longer than about 10 ps, which is not useful for measuring the sub-picosecond compressed pulses in either UEM or UED. However, besides having possible applications in systems using longer duration pulses, it will be shown that the DRC can still be (and is) used in the UIC project to determine whether the CRC EM field is at the proper phase for compression when a pulse arrives. Moreover, the development of the DRC lays the groundwork for higher frequency pulse detection cavities that could potentially enter the sub-picosecond time regime. In Chapter 2, I will provide an overview of all background theory relevant to the application of the RF resonant cavities in a UEM system. The major topics covered include the theory of cylindrical waveguides and cavities (which are simply waveguides with end boundaries), and the nature of resonant eigenmodes in ideal lossless cavities. This is followed by a discussion of power loss in real cavities. Next, the theory of RF power delivery to a resonant cavity is detailed. This includes basic transmission line theory and coaxial cable EM field theory, inductive coupling and eigenmode stimulation by current sources, and impedance matching of the power source to the cavity to prevent reflection of power back to the source. Chapter 3 begins by outlining the main design considerations for the CRC, and then provides a simplified model of the operating principle. Next, it is shown how all design criteria are met, and the details of the construction and characterization of the CRC follow. Considerable time is then spent covering the theory and operating principle of the DRC. A unique method for testing the 4 DRC’s ability to detect changes in larger pulse durations is then presented. It is shown that a large duration electron pulse, for example, an 80 ps pulse (which is too long to be generated by the UIC system), can be simulated by two shorter (~4 ps) electron pulses separated by 80 ps. It is also shown that this double pulse method can be used to calibrate the DRC for direct long pulse duration measurement. Chapter 3 ends by describing the process of synchronizing the pulsed laser and the CRC, along with an analysis of system stability. In Chapter 4, I present the results of experiments performed to test the theory and construction of the CRC and DRC. There are two main experimental results, the first being the successful calibration of the DRC using the double pulse method. This type of experiment has not been performed before. Then, results of a compression experiment using double pulses show the CRC is properly synchronized, and achieves successful compression. The calibration of the DRC is also shown to be accurate. The thesis concludes with Chapter 5 in which a summary is given and future prospects are discussed. 5 2 Theory 2.1 Electrodynamics of Resonant Cavities The goal of this section is to provide a basic understanding of how EM fields behave inside a resonant cavity, specifically a right circular cylindrical cavity (RCCC). Figure 2-1 shows the basic geometry of the problem. Figure 2-1 RCCC Geometry Relevant coordinates and origin to be used throughout this document when discussing cylindrical cavities. Initially, in our analysis, we will be seeking exact solutions to Maxwell’s equations for cavities with perfect boundary conditions and material properties, i.e. perfectly smooth metallic boundaries with infinite conductivity and lossless linear dielectrics inside the cavity. The actual fields are of course only approximated by these exact solutions, and perturbation (or variational) methods are used when more accurate solutions are needed for real-world cavities. In our experiments with electron beams, we will be using a RCCC that has an axial hole for the beam to pass through, an aperture and slot for the power coupling loop to be inserted, a non-zero skin depth at the metallic boundaries, and a dielectric with 6 a non-zero loss tangent. The effect of these real-world cavity imperfections on the exact field solutions will be discussed. We will find that the exact solutions work well in the majority of situations and can therefore be used in estimates for resonant frequencies, power consumption, energy transferred to the electron pulses, DRC signal power, etc. Consequently, it is important to analyze these exact solutions in some detail. There are several main results from this analysis which will be summarized here. First, a cylindrical cavity can be thought of as a cylindrical waveguide with closed ends, so solutions to the waveguide problem that meet the end boundary conditions are the solutions to the cavity problem. Second, the closed ends create a completely enclosed volume which leads to a discrete spectrum of resonant frequencies and corresponding field modes. Third, because of the geometry of waveguides and cavities, we can assume a sinusoidal dependence of the fields in the axial direction. As will be shown, this leads to the overall problem reducing to that of finding a scalar function which represents each mode. The complete vector modes and fields can then be found by taking vector derivatives of this scalar function. Finally, the modes naturally separate into two distinct types, transverse magnetic (TM) and transverse electric (TE). TM modes have no axial magnetic field component, and TE modes similarly have no axial electric field component. Any general field inside a resonant cavity can be expressed as a linear combination of TM and TE modes. A specific TM mode, suited for compression of an electron beam traveling along the axis of the cavity, will be used in applications relevant to this document. 2.1.1 Complex Notation for Time-Harmonic Fields The EM fields in the following analysis will be assumed to have sinusoidal (harmonic) time dependence. It is much simpler to deal with such fields using complex notation and so a brief review of this notation is given here. The following are Maxwell’s equations in complex form for time-harmonic fields: 7 E i H ˆ )E J s . H ( i It is standard to use H instead of B 2.1 for waveguide and cavity theory (it allows for a simple analogy between waveguides and transmission lines). Here, Js represents what will be referred to as a source current that is applied to a system and elicits a response from the system. This response can itself include currents which are accounted for in other terms in the equations. All vector components in the above equations are phasors, and include the phase and amplitude information about each component of the fields. For example, the phasor Ez (r ) Ez (r ) e i where is a phase constant. It is important to note that phasors are a function of position only. The relationship between a phasor quantity and an instantaneous real field quantity will be defined as the following: Ez (r , t ) 2 Re( Ez (r )eit ) 2 12 Ez (r )eit Ez* (r )eit , where the factor of 2 means that the magnitude of a phasor is taken to be the root mean square (rms) value of the real time-harmonic quantity it represents (simply a convention). In Equation 2.1 we take and to be real and we allow ˆ to be complex, ˆ i , where the imaginary part will account for dielectric loss (the negative sign is again a convention). All of these quantities can be a function of frequency. Oftentimes when dealing with harmonic fields, instantaneous quantities related to the fields will not be of interest; instead time-averaged quantities are more useful. The most common quantity of this type (and one which will be analyzed in detail in this document) is EM field power. For example, the expression Ez (r , t ) Jz (r , t ) can be shown to represent instantaneous power flow 8 density in the z-direction (here Ez ( r , t ) represents the z-component of the electric field, and Jz ( r , t ) represents the z-component of current density). The timeaverage value of this quantity is of interest for the harmonic fields that will be used in the analysis. Time-average quantities will be represented by the following over-bar notation: B ( r , t ) , for some quantity B . So time-average power density is represented by the expression Ez (r , t ) Jz (r , t ) . It will be useful to represent this type of expression (a time-average of a product of two harmonic field quantities) in the complex phasor notation described above. From the definition of the complex phasor it is evident that Ez (r , t ) Jz (r , t ) 2 12 Ez (r )eit Ez* (r )eit 2 12 J z (r )e it J z * (r )eit . After multiplying out all the terms and rearranging, it can be shown that the expression on the right-hand side is the real part of another expression, and therefore Ez (r , t ) Jz (r , t ) 2 Re 12 Ez (r ) J z* (r ) Ez (r ) J z (r )e 2it . It is seen that the time-average of the second term on the right side is simply zero, so the final expression for the time-average power is Ez (r , t ) Jz (r , t ) Re Ez (r ) J z * (r ) . 2.2 Thus, the real part of the product of one complex phasor and the conjugate of another is equal to the time-average value of the product of the two time-harmonic field quantities. 9 In the remainder of this text, when a field is referred to (electric field, magnetic field, EM field, etc.), it will be assumed that the field quantity in question is a phasor quantity unless otherwise specified. In certain cases the term “phasor” will be included, to avoid confusion when the distinction between phasor quantities and time-dependent quantities is especially important (or possibly unclear). Time-dependent quantities will always be written as E (for the electric field, for example) and the corresponding phasor E. 2.1.2 Cylindrical Waveguide Field Solutions In general, an EM waveguide can be thought of as any device which allows for the propagation of EM waves along a directed path. In this section, it will be assumed that a waveguide is filled with a homogenous lossless linear dielectric, has perfect conducting boundaries, a constant cross-section, and allows propagation in a straight line along the z-direction, which will always be aligned with the central axis of the guide. The transverse plane will be defined as the plane perpendicular to the z-direction, and the transverse field will be defined as the field (electric, magnetic, etc.) made up only of components aligned in the transverse plane. It is assumed that there are no current or charge sources anywhere inside the waveguide. There are also no end boundary conditions, so that the guide is taken to be infinitely long. Maxwell’s equations inside the guide then have the form: (a) E i H (b ) E (c ) H i E (d ) H 0 0. 2.3 A wave equation for the fields can be derived in the usual way and takes the form: 10 E ( 2 2 ) 0. H 2.4 The assumption about the sinusoidal dependence of the fields in the z-direction is now applied. Using the geometry of Figure 2-1, the fields can then be expressed in the following form: E ( x, y, z, t ) E ( x, y ) ikz it , e H ( x , y , z , t ) H ( x , y ) 2.5 where k can be complex, and the stands for waves traveling in the positive (+) z-direction or negative (-) z-direction. Having made this assumption about the z-direction dependence of the fields, it can now be shown that all field components for a particular waveguide mode can be derived from a single scalar function that represents that mode. The first step is to separate Maxwell’s vector Equations 2.3(a,c) into z-component equations and transverse component equations. It is useful to define two operators; the transverse gradient vector operator t zˆ z , and the transverse Laplacian scalar operator t2 2 2 z 2 . Also, the expressions Et and H t will be used to represent the transverse components of the fields. To get the separated equations, the cross product and dot product of the unit vector ẑ is taken with each of Maxwell’s curl equations. For example, for 2.3(a) we take zˆ ( E ) zˆ (i H ) and zˆ ( E ) zˆ (i H ) , and similarly for 2.3(c). Maxwell’s divergence equations can also be rewritten, separating the transverse part, resulting in (using our specified dependence in the z-direction) 11 (a) ikEt i ( zˆ H t ) t Ez (b ) zˆ (t Et ) i H z (c ) ikH t i ( zˆ Et ) t H z (d ) zˆ (t H t ) i Ez 2.6 (e) t Et (f) t H t Ez z H z , z where again the stands for waves traveling in the positive (+) z-direction or negative (-) z-direction. One can now observe that if the z-components of the fields are known, the other components can be computed from the above equations. If one simultaneously solves Equations 2.6(a) and 2.6(c) for Et and H t , the result is (a) (b ) i k (t Ez ) ( zˆ t H z ) ( 2 k 2 ) i Ht k (t H z ) ( zˆ t Ez ), ( 2 k 2 ) Et 2.7 which shows the explicit relationship. It is important to note that these equations are only valid if at least one of the z-components (Ez or Bz) is non-zero. (If both are zero there can still exist what is called a TEM mode, to be discussed in a later section). At this point a solvable equation is needed for the scalar z-components alone. This is accomplished by substituting the expressions on the right-hand side of Equations 2.5 into Equation 2.4, and then isolating the z-component parts. This results in: E t2 ( 2 k 2 ) z 0, H z 12 2.8 which is the desired equation. Next, boundary conditions specifically for the zcomponents are needed. The general boundary conditions for time-varying fields at the surface S of a perfect conductor are (a) nˆ E 0 (b ) nˆ H 0 (c ) nˆ E f (d ) nˆ H K f , S S S 2.9 S where n̂ is a unit normal vector at the surface, f is the free surface charge, and K f is the free surface current. As one generally does not have knowledge of the surface charge or surface current, only the general boundary conditions in 2.9(a) and 2.9(b) will be used. From Condition (a) it is clear that the boundary condition for Ez must be E z S 0. 2.10 The boundary condition for Hz is not as obvious. If one takes the dot product of the unit normal n̂ with Equation 2.6(c) at the boundary surface S, one obtains H ik (nˆ H t ) i nˆ ( zˆ Et ) z , S S n S 2.11 where n is the normal derivative at a point on the surface. The first term on the left is zero because of the general boundary Condition 2.9(b). The second term is zero because of general boundary Condition 2.9(a). What remains is the boundary condition for Hz : 13 H z n 0. 2.12 S One now has equations and boundary conditions for Ez and Hz, from which the complete vector fields can be solved. The problem can be simplified further, however, if the solutions are first divided into two types. For the first type, one sets H z 0 everywhere, which automatically satisfies boundary Condition 2.12 and Condition 2.10 remains. These are called transverse magnetic or TM modes, since the magnetic field only has transverse components. For the second type, one sets E z 0 everywhere, satisfying boundary Condition 2.10 and leaving Condition 2.12. These are the transverse electric or TE modes. For TM/TE modes, Equations 2.7 simplify nicely, becoming for TM modes (a) (b ) i (TM ) k (t Ez ) ( 2 k 2 ) i Ht ( zˆ t Ez ) (TM ), ( 2 k 2 ) Et 2.13 and for TE modes (a) (b ) i ( zˆ t H z ) (TE ) ( 2 k 2 ) i Ht (TE ). k (t H z ) ( 2 k 2 ) Et 2.14 From these equations it is clear that for TM(TE) modes, knowledge of a single scalar function, namely Ez(Hz), that satisfies wave Equation 2.8 and the TM(TE) boundary condition, will allow calculation of the full vector fields. For TM 14 ikz it , modes Ez can be represented as a scalar function TM ( x, y ) multiplied by e where TM ( x, y ) is a function of the transverse independent variables x and y only. Similarly, for TE modes one has H z TE ( x, y)e ikz it , with TE ( x, y) being called the scalar TE mode function. The mode functions will still solve the wave equation TM ( x, y ) t2 ( 2 k 2 ) TE 0, ( x, y ) 2.15 and when combined with the TM(TE) boundary Conditions 2.10(2.12) one has a unique eigenvalue problem for ψTM(ψTE). The wave equation is the same, but the different boundary conditions mean the two sets of solutions will have different eigenvalues. 2.1.3 Circular Cylindrical Cavity Field Solutions If a cylindrical waveguide has closed, perfectly conducting ends (the ends being in the transverse plane), then the waveguide becomes a cylindrical cavity. Everything about the problem remains the same, except for the ends adding an additional boundary condition. To meet this boundary condition, for each mode a wave traveling in the positive and negative direction will need to be combined to create standing waves inside the cavity. The individual traveling waves remain identical to our waveguide solutions. It will be assumed in advance via symmetry that the contribution from the forward and backward traveling waves is of the same magnitude. The solution will be developed for TM modes in a cavity, noting that the TE mode analysis would be analogous. For any TM mode the general solution in a cavity will be 15 Et Et Et Ez A Ez A Ez H H H t t t TM cavity , 2.16 where the +(-) sign refers to a wave propagating in the positive(negative) zdirection, and it has been assumed an equal contribution from each wave (note there is no Hz for the TM modes). Therefore, one simply needs to determine the constant A which allows the end boundary conditions to be met. Consider first Et A( Et Et ); the waveguide solutions for the TM fields 2.13 allow this equation to be expressed as Et ik ( k ) 2 2 t TM ( x, y)eit ( Aeikz Aeikz ), where Ez TM ( x, y)eikz it has been used. 2.17 To meet the end boundary conditions, Et must be zero at z 0 and z d (see Figure 2-1) so Et must have a sin(kz ) dependence with k p d , p 0,1,2... . This can be accomplished by setting A 1 2 . Using this value for A also fully determines the z-dependence for Ez and H t , which have expansions analogous to Equation 2.17. Setting 2 ( 2 k 2 ) and with k p d , the final expressions for the TM cavity fields become: 16 Et p p z TM sin t ( x, y ) 2 d d (TM cavity ) p z Ez TM ( x, y )cos d i p z TM H t 2 cos zˆ t ( x, y ) d (TM cavity ) 2.18 (TM cavity ). For the TE fields, the boundary condition requires Hz to equal zero at z 0 and zd, so it is Hz that will need the sin(kz ) dependence with k p d , p 0,1,2... . The expressions for the TE cavity fields then become p z TE sin zˆ t ( x, y ) (TEcavity ) d p p z TE H t 2 cos (TEcavity ) t ( x, y ) d d Et i 2 p z H z TE ( x, y )sin d 2.19 (TEcavity ). The most important thing to note from these two sets of equations is that for any particular mode, we only need to solve for a single scalar mode function in order to derive all of the cavity fields, just like in the waveguide problem. At this point we can finally investigate the nature of these scalar mode functions. Consider the TM functions TM ( x, y ) , which solve Equation 2.15 subject to boundary Condition 2.10, with k p d and Ez TM ( x, y)eikz it . One has: 2 t 2 TM ( x, y) 0, TM ( x, y) 0, S 17 2.20 which is simply the two-dimensional scalar Helmholtz equation with Dirichlet boundary conditions, and eigenvalues given by 2 2 p d . 2 2.21 The solutions to this problem are well known. In cylindrical coordinates, which TM will be used for the RCCC, the boundary condition becomes ( , ) R 0 (see Figure 2-1) and the solutions are TM mn ( , ) E0 J m ( mn )eim , 2.22 where E0 is a normalization phasor, mn xmn R , and xmn is the nth root of the Bessel function J m ( x) . The integer m can take the values m 0,1,2,... , and the integer n can take the values n 1,2,3,... . Solving Equation 2.21 for the TM eigenfrequencies ωTM generates TM mnp 1 2 xmn p 2 2 2 R2 d (TM ). 2.23 The corresponding TM modes are labeled TMmnp. For TM modes, the integer p can take on values p 0,1,2,... in order to meet the end boundary conditions. In the experiments relevant to this document, the mode of greatest interest is the m 0 , n 1, TM p 0 mode, or TM010, with eigenfrequency 010 . As x01 2.405 , the frequency for this mode becomes 18 TM 010 2.405 . R 2.24 TM ( , ) E0 J 0 2.405 R , and using The scalar mode function simplifies to 01 Equations 2.18 one can solve for the fields: 2.405 it Ez E0 J 0 e R H i 2.405 it E0 J1 e R TM 010 2.25 TM 010 . Finding the TE modes and frequencies proceeds in the same manner, by solving TE for mnp ( , ) using the scalar Helmholtz equation with the appropriate boundary condition, and then using Equation 2.19 to find the vector fields. The expression for the eigenfrequencies becomes TE mnp 1 2 xmn p 2 2 2 R2 d (TE ), 2.26 2 is the nth root of the derivative of the mth Bessel function J m ( x) . where xmn Also, for TE modes the integer p cannot be zero but must start at 1. The other integers m and n take the same values as TM modes. The field pattern for the experimentally employed TM010 mode (Equations 2.25) is shown in Figure 2-2. From this pattern, it is clear that n̂ H is nonzero at all surfaces of the cavity. Going back to the boundary conditions for a perfect conductor (Equations 2.9), it is evident from Condition (d) that there must be a surface current flowing on all surfaces of the cavity, and this surface current is 19 equal to n̂ H at each point on each surface (due to Condition (c) there must also be a surface charge on the end surfaces of the cavity, but this is less important). Figure 2-2 TM010 Mode Pattern Cylindrical resonant cavity TM010 eigenmode pattern, with the H-field lines shown in blue and the E-field lines in red. A sinusoidal time-dependence is assumed. The flow pattern for the surface current is shown in Figure 2-3. It is important to note that when the boundaries are not perfect conductors but have finite conductivity, these currents will no longer be surface currents but will be volume current densities that penetrate a certain distance (called the skin depth) into the boundary material (copper, for example). These current densities will cause ohmic losses in the boundary material and they are a major source of power loss when operating a resonant cavity. This will be explored in detail in the next section. 20 Figure 2-3 TM010 Mode Current Pattern Surface current pattern for the TM010 mode. The current is required to meet the boundary conditions for the cavity. 2.1.4 Cavity Power Loss In UEM, the usual purpose of a resonant cavity is to affect the temporal dynamics of an electron pulse that passes through it. The cavity accomplishes this task by generating strong internal EM fields that then interact with the electrons in the pulse. One of the goals of resonant cavity design is to make sure any power that is delivered to the cavity is efficiently transformed into useful energy in the stimulated field mode(s) that may then be transferred to the electrons. It is therefore desirable to minimize all sources of power loss in the system that deter from this goal. The method that will be used to investigate power flow and loss in cavities is the well-known Poynting theorem, in complex form for fields with sinusoidal time-dependence. To derive the relevant equation for our specific purposes, one starts with the complex Equations 2.1, multiplying the first equation by H * and the complex conjugate of the second equation by E. The point of doing this is to create a term (in the second equation) that has the form E J s* , the real part of which will be related to time-average power flow 21 supplied to a system by its sources, which is what will be of interest. The result of the multiplication is H * E i H * H ˆ E E * E J s*. E H * i Subtracting the first equation from the second, and letting ˆ i , one obtains 2 2 2 E H * H * E E E i H E 2 EJ . s* The left-hand side of this equation can be simplified using a vector identity, giving 2 2 2 ( E H * ) E E i H E 2 EJ . s* The complex Poynting vector will be defined as S E H , and note that E * 2 2 and H are time-average quantities, since they are technically the real parts of expressions involving a phasor multiplied by the complex conjugate of another phasor. One can now integrate the equation over a volume V that is enclosed by a surface S. A unit vector n̂ is defined to be normal to S and pointing out of S, away from the volume V. The divergence theorem is then applied to the left-hand side of the equation. This results in: S dS (nˆ ) E J s* E 2 E 2 i H 2 E 2 dV . 22 2.27 Rearranging the terms and defining the following quantities: Ps E J s*dV S dS (nˆ ) P E dV P E dV W E dV W H dV Pf 2 d, 2 d , e 1 2 m 1 2 2 2 Complex Power Supplied by Sources Complex Power Flowing Out of S Time AverageConduction Loss in V 2.28 Time Average Dielectric Loss in V Time Average Stored Electrical Energy in V Time Average Stored Magnetic Energy in V , allows Equation 2.27 to be rewritten in the much simpler form Ps Pf Pd , Pd , 2i(Wm We ). 2.29 The real part of this equation is Ps Pf Pd , Pd , , 2.30 where it is noted from the definitions of Ps and Pf that their real parts have the form of a time-average quantity. In words, Equation 2.30 states that the timeaverage power supplied by the sources of a system (in our case a cavity system) is equal to the time-average power flowing out of the surface of the system, plus the time-average power dissipated within inside the system due to conduction and dielectric losses. The imaginary part of Equation 2.29 is Im Ps Im Pf 2 (Wm We ). 23 2.31 This is an equation that describes the behavior of stored energy that does not dissipate. Inside the system, energy stored in the fields switches from electric form to magnetic form and back during every cycle. If there is an excess of one form over the other on average, Equation 2.31 tells us that part of this excess either is sent back to the source and returns to the system every cycle, or flows out of the surface and returns to the system every cycle. The part that is sent back to the source for temporary storage each cycle is Im( Ps ) . The part that flows out of the surface for temporary storage is Im( Pf ) . A system with no sources supplying power and no power flowing through the surface will have an equal amount of time-averaged stored electric and magnetic energy. For example, if one calculated the time-average stored energies in the exact cavity solutions described earlier in Section 2.1.3 (operating at their respective eigenfrequencies), no energy imbalance would be found, since these solutions assumed the cavity was source-free with perfectly conducting boundaries allowing no power to flow through them. One of the main sources of power loss is ohmic loss due to currents flowing in the conducting cavity boundaries. For real cavities these boundaries do not have perfect conductivity, which was assumed previously to simplify boundary conditions when solving for the exact field solutions. These exact solutions are still useful, however, since the boundary materials (copper, silver, etc.) used in practice have very high conductivity, and the change to real boundary conditions can be viewed as a small perturbation. The exact field solutions can therefore be used to approximate the power loss to lowest order, which is generally sufficient for these materials. Interestingly, in order to determine the power lost to boundary currents, one does not need to calculate the currents directly, but can instead approximate the (perturbed) EM fields at the boundary, on the cavity (dielectric) side, and then 24 integrate the real part of the resulting complex Poynting vector over the cavity surface. According to the theory developed in the previous section, this integral will give the time-averaged power flowing out of the surface of the cavity. This must be equal to the power lost by currents in the conducting material since there is no other place for the power to go. To determine (approximately) the EM fields at the boundary on the cavity side, one must actually start with Maxwell’s equations in complex form inside the conducting material surrounding the cavity. These equations will be applied at the boundary, but on the conductor side, and then boundary conditions will be used to show these fields must be continuous across the boundary into the cavity. The equations inside the conductor are H c i E c E c E c i H c , where the ‘c’ superscript means these are the fields inside the conductor. For materials like copper at radio frequencies, , so the first term on the right- hand side of the first equation can be neglected, which after rearranging gives Ec Hc 1 Hc i c Ec. Next, it is noted that EM fields in a good conductor decay very quickly in the direction normal to the surface of the conductor. Mathematically, this means that the normal derivative of the fields will be much greater than the transverse derivatives. If one defines a coordinate that equals zero on the surface of the conductor and increases in the normal direction to the surface (in the direction of 25 the surface unit vector n̂ , away from the volume), then mathematically one can write t and let n̂ in the previous set of equations. This results in: c H i c Hc nˆ E. c 1 Ec nˆ 2.32 These coupled equations can be combined into a wave equation and solved, giving solutions of the form [10] H ( ) H e e c c E c ( ) 1 i i (i 1)(nˆ H c )e e 2.33 1 2 , c 2 where Equations 2.32 have been used to solve for the electric field in terms of the magnetic field. The additional subscript on the fields means, in this approximation, they are everywhere parallel to the boundary surface S and have no component along n̂ . The parameter , called the skin depth, has been defined, and is equal to the distance inside the conductor at which the field amplitude decays to 1 e of its value at the conductor surface. Notice that the higher the frequency and conductivity, the smaller the skin depth, which is basically a measure of how deep EM field penetrate into a conducting surface. The only unknown in Equations 2.33 is the value of H c , the parallel component of the magnetic field on the conductor side of the boundary, where 26 the coordinate is equal to zero. According to Ohm’s Law, a current density equal to J is E inside a conductor, and this may not be formulated in terms of an infinite surface current density as this would require an infinite electric field at the surface for a finite conductivity. Consequently, there can be no surface current along the surface S separating the cavity volume from the conductor. As only a surface current can cause a discontinuity in the parallel component of a magnetic field across a boundary, the magnetic field must be continuous across S; that is Hc H , where the superscript has been removed to indicate we are referring to the parallel component of the magnetic field at the boundary but on the cavity (dielectric) side. At this point, the solutions presented in Section 2.1.3, where infinite conductivity was assumed, can be employed. Since the actual conductivity is not infinite but still very large, as an approximation it will be assumed, for any operating mode of the cavity, that the parallel component of the magnetic field at the surface (on the cavity side) is the same as stated in the prior solutions. As this component is continuous across the boundary, it can be used in Equations 2.33, which apply in the conductor, to calculate the parallel component of the electric field at the boundary on the conductor side. Parallel components of the electric field are always continuous across a boundary, so this value will be the same on the cavity side of the boundary. Thus, values for the parallel components of both the electric and magnetic fields at the boundary on the cavity side are known. Using these values of the fields, the expression for time-average power flowing out of the surface S surrounding the cavity is Pf Re S dS (nˆ) Re E H dS (nˆ) 4 H 27 c 2 dS , 2.34 where H , as previously stated, is the component of the magnetic field parallel to the surface S on the cavity side, and taken from the eigenmode solutions of Section 2.1.3 as an approximation. The other main source of power loss inside the cavity is dielectric loss. For the eigenmode solutions, in addition to assuming perfectly conducting boundaries, it was also assumed that the dielectric material inside the cavity was completely lossless; that is, 0 . For real cavities this is not true, although the value of will be very small for good dielectrics such as fused silica and YAG. This fact will again allow the eigenmode solutions to be used as an approximation, since the actual field will not be altered significantly by the presence of relatively small loss. As a result, the electric field of an eigenmode solution in Equation 2.28 is used to calculate the term ( Pd , ) for dielectric loss within the cavity volume. Note that Equation 2.28 also contains a term for conduction loss within the cavity (Pd , ) . As will be discussed in Section 2.2.3, an operating cavity will include a small internal current carrying loop of wire that delivers power to the cavity, and the wire will have conduction losses. This loss is also small and the eigenmode solution can still be used for calculations. 2.2 Power Supply So far, discussion has included the EM field modes that can exist inside a RCCC and also the power losses associated with operation of those modes, but how these modes are excited and driven has not been described. Power must be delivered from a source to the RCCC in order to drive the desired eigenmode to the proper amplitude. Once the proper amplitude is reached, power will be continuously needed to maintain that amplitude because of cavity losses. In the experiments described in this document, power from the source is delivered to a RCCC along a coaxial cable. A coaxial cable can be considered a special type of metallic waveguide that has an inner and an outer conducting surface, with the 28 volume in between capable of sustaining EM waves similar to those in the hollow waveguide already discussed. The major difference is that the inner conductor in a coaxial cable allows the existence of a TEM (transverse electromagnetic) wave that is not possible in a hollow guide. A TEM wave has no field component (electric or magnetic) in the axial direction and, as will be shown, this property qualifies the coaxial cable as a type of transmission line. There are several advantages to using transmission lines to transport power, the main advantage being that the characteristic impedance of the line is independent of frequency. This allows the line to be compatible with equipment (such as power supplies, connectors, couplers, etc.) that operate over a range of frequencies and are designed for use with lines having standardized characteristic impedances. This is extremely important, since any mismatch between the impedance of the line and the effective impedance of the equipment it is attached to will cause reflections of power back down the line, away from its intended destination. This phenomenon is analogous to an EM plane wave being partially reflected when it encounters a medium with different dielectric properties. Once power has traveled from the source to the cavity via transmission line, the proper eigenmode of the cavity is then, in our case, stimulated inside the cavity by a technique called inductive coupling (other techniques are used in different applications). In order to prevent any reflection of power back down the line away from the cavity, the combined effective impedance of the cavity and inductive coupling system must match the characteristic impedance of the line. This is the art-form known as impedance matching. The theory behind these concepts is described in this section. 29 2.2.1 Transmission Lines and the Coaxial Cable TEM Mode Figure 2-4 Transmission Line Concept Parallel wire transmission line model used for derivation of the transmission line equations. There are many different kinds of transmission lines used to deliver power; the parallel wire transmission line is shown in Figure 2-4 and is typically used to demonstrate the basic concepts. Here, one considers the positive z-direction to be the direction pointing along the line towards the load. If a sinusoidal alternating voltage is applied by a source to one end of the line, voltage and current waves will travel down the line carrying power with them. We are concerned with a particular type of wave or mode that can be sustained on a transmission line, called the transverse electromagnetic (TEM) mode or simply a transmission line mode. For this type of mode, at any particular position or crosssection along the line, there will be a uniquely defined voltage and current related to the transverse electric and magnetic fields surrounding the wires (again, the fields have no z-components for the TEM mode, only transverse). For the parallel wire set-up, the unique definitions are 30 V ( z ) E dl I ( z ) H dl , C1 C2 2.35 where C1 is a path connecting the two wires at the particular cross-section and C2 is a path surrounding one of the wires at the cross-section. The values must be unique regardless of the specific chosen path in order for the mode to be considered a transmission line mode. These voltage and current values will change as we move in the z-direction. The fundamental postulate for transmission lines states that the change in voltage is proportional to the current at any point, and conversely the change in current is proportional to the voltage. From this postulate and Figure 2-4, one can generate the well-known transmission line equations dV ( z ) ZI ( z ) dz dI ( z ) YV ( z ). dz 2.36 Here the proportionality constants are labeled Z, the impedance per unit length of the line, and Y, the admittance per unit length of the line. These constants, which can have many different forms depending on the type of transmission line, will be assumed to be independent of z for our purposes and they are not phasor quantities. V(z) and I(z) are phasor quantities, with a sinusoidal time-dependence assumed for the real voltage and current values. One can derive wave equations for the voltage and current phasors from the transmission line equations and obtain d 2V ( z ) ( ZY )V ( z ) 0 dz 2 d 2 I ( z) ( ZY ) I ( z ) 0. dz 2 2.37 These standard one-dimensional Helmholtz equations have the general solutions 31 V ( z ) V0e ZY z ZY z I ( z) I0 e V0e ZY z ZY z I0 e 2.38 , where we see that waves traveling in the positive or negative direction are allowed and have a sinusoidal z-direction dependence. Suppose only a wave traveling in the positive direction is present ( V0 I 0 0 ). If one substitutes the resulting solutions from Equations 2.38 into Equations 2.36 and divides, the ratio of the voltage and current phasors is found to be V0 Z Z0 , I0 Y 2.39 where Z0 is defined to be the characteristic impedance of the line. For a wave traveling in the negative direction only, one obtains V0 Z Z0 . I0 Y 2.40 As long as there is only a single wave present, traveling in one direction, the absolute value of the ratio of voltage to current will be equal to Z0 for the entire length of the line and be independent of the z-coordinate. When waves traveling in both directions are present, the ratio of the total voltage and current at any particular cross-section becomes a function of the position on the line, and this ratio Z(z) is called the impedance of the line. One can define the complex power traveling down the line as 32 Pf ( z ) V ( z) I * ( z ), 2.41 where, as usual, the real part represents time-average power flow. For a lossless transmission line, the time-average power flow will be independent of the zcoordinate, but for real lines there will be power losses and an attenuation per unit length can be defined for the line. In this case the quantity ZY will be complex, with the real part being related to the attenuation. As previously stated, a coaxial cable can sustain a TEM wave and therefore can potentially be treated as a transmission line. From Equations 2.6 (b) and (e), if Ez and Hz are set equal to zero (TEM), one finds that t Et t Et 0 , and it follows that t2 Et 0 . It is also known that the tangential electric field must be zero at the inner and outer boundary surfaces, since they are assumed to be (near) perfect conductors. So for any particular cross-section of the cable, it is evident that the boundary value problem for the transverse electric field is identical to the two-dimensional electrostatic boundary value problem. A similar analysis of the magnetic field shows that they are solutions to the two-dimensional magnetostatic boundary value problem. Both of these problems of course have well-known solutions [11], whose patterns are shown in Figure 2-5. As in the case of the RCCC, boundary conditions for these magnetic field patterns require surface currents to flow on the inner and outer conductor surfaces, and these currents are also shown in the figure. The current on the inner conductor will be key to coupling the coaxial cable to the resonant cavity and delivering power, as will be discussed later. 33 Figure 2-5 Coaxial Cable Geometry Surface current flow and TEM mode pattern for a coaxial cable. Also shown are the integration paths used to calculate equivalent transmission line voltages and currents. It is now clear why a TEM mode cannot exist in a hollow waveguide; the electrostatic solution for the electric field would be zero everywhere without the inner conductor present. It is important to remember that although the patterns for the field (phasors) look identical to a static field (phasor) solution, the actual electric field still carries with it a sinusoidal time-dependence (the phasors only contain information about the amplitude and phase of the true time-dependent fields) as well as the usual assumed z-direction dependence. The solutions for the fields (phasors) are E ( z ) E0 ikz e 2 r H ( z ) H 0 ikz e , 2 r 2.42 where E0 and H0 are normalization phasors. The transverse electric and magnetic fields ( E ( z ) and H ( z ) ) of the coaxial cable TEM mode can of course be related to each other; from Equation 2.6 (c), with Hz set equal to zero, one finds H ( z ) ( k ) E ( z ) . From Equation 2.8, recalling that t2 Et 0 , the result k is obtained. Inserting this value of k into the transverse magnetic field expression results in 34 H ( z ) E ( z ), 2.43 where the (+) stands for waves traveling in the positive z-direction and the (-) represents waves traveling in the negative z-direction. It is now possible to express both the electric and magnetic fields in terms of the electric normalization phasor E0, which will turn out to be convenient later: E ( z ) E0 i e 2 z H ( z ) E0 i e 2 z . 2.44 In order to represent the coaxial cable as a transmission line, it is now necessary to define the unique voltage and current phasors at a cross-section of the line in terms of the transverse fields (recall that these unique phasor definitions must exist for the cable to be considered a transmission line). One would like to do this in such a way that the total complex power traveling down the cable will be * ( z ) , in analogy to the transmission line complex power equal to Vcoax ( z ) I coax Equation 2.41. To find the total complex power flowing down the cable, the complex Poynting vector is integrated over a cross-section of the cable. For example, for a wave traveling in the positive z-direction we have Pfcoax ( z ) S ( z) dA E ( z ) H ( z ) dA[( ˆ ˆ ) zˆ] * A A E0 E0* E0 b 2 d ln . 2 2 2 a a 2 b 35 2.45 (Note that the expression here for the complex power only has a (z-independent) real part. This will not be true when waves traveling in both directions are present.) If we make the following definitions, where the integration paths C1 and C2 are shown in Figure 2-5: E0e i z E Vcoax ( z ) E ( z ) d 0 ln b a e i 2 2 C1 a b I coax ( z ) H ( z ) C2 2 0 i z E0e 2 d z i Ee 0 2.46 z , then the desired result for complex power can be obtained. It is important to note that, because of the nature of the TEM field pattern, the voltage and current values will be independent of the integration paths as long as C1 connects the inner and outer conductors and C2 encircles the inner conductor. This is not in general true for TM and TE modes. For those modes any type of transmission line analogy will therefore be an approximation, but for the TEM mode the analogy is exact and much more useful in predicting the behavior of the power flow in the coaxial cable. To complete the transmission line analogy it is necessary to show that Vcoax and Icoax, as they have been defined, satisfy the transmission line Equations 2.36. Inserting the above definitions into Equations 2.36 we obtain dVcoax ( z ) E0e i i ln b a dz 2 dI coax ( z ) i E0e i z . dz If the following definitions are made: 36 z 2.47 Z coax i ln b a 2 Ycoax i 2 , ln b a 2.48 then Equations 2.47 will take the form dVcoax ( z ) Z coax I coax ( z ) dz dI coax ( z ) YcoaxVcoax ( z ), dz 2.49 in exact analogy to the transmission line equations. One can also define the characteristic impedance for the coaxial cable in analogy with Equations 2.39 and 2.40: Z 0coax V0coax I 0coax V0coax I 0coax Z coax ln b a . Ycoax 2 2.50 In analogy to the case of the parallel wire transmission line, Equations 2.49 will have solutions with the general form ZcoaxYcoax z V0coax e ZcoaxYcoax z 0coax Vcoax ( z ) V0coax e I coax ( z ) I 0coax e I e Z coaxYcoax z Z coaxYcoax z 2.51 . As before, if only a positive or negative traveling wave is present the impedance of the cable, Z coax ( z ) Vcoax ( z ) , I coax ( z ) 37 2.52 will be equal to the characteristic impedance of the cable. If waves in both directions are present, the impedance becomes a function of the z-position on the line. 2.2.2 General Impedance Matching The general setup for what is known as a low-frequency network is shown in Figure 2-6. The thick black lines represent the transmission line portion of the network. In reality this could be any type of transmission line, in our case it represents the coaxial cable. Attached to the beginning of the line is the power source which generates the voltage and current waves traveling down the line. It is characterized by its impedance labeled ZS. This means that at the cross-section where the source is attached to the line ( z s ), the ratio of the voltage to the current, as defined previously, will be equal to ZS (and similarly for all impedances we define). A reference plane along the line is defined arbitrarily and the z-coordinate is measured from this plane. ZIN is defined as the (input) impedance at the reference plane and can be considered the impedance the source would “see” if it were hooked up at the reference plane ( z 0) . The end of the transmission line is hooked up to the load, located at z l , whose impedance is denoted by ZL. The transmission line itself has the characteristic impedance Z0. A TEM wave traveling in the positive z-direction, towards the load, is referred to as the incident wave. A TEM wave traveling back towards the source in the negative z-direction is referred to as the reflected wave. 38 Figure 2-6 Low Frequency Network Basic components of a low frequency network, showing the source, load, and reference plane positions along a transmission line. It is important to understand what is meant by low-frequency. Recall that for a coaxial cable, a unique voltage and current, and therefore a unique impedance, could be defined at a particular line cross-section at any frequency due to the nature of the TEM mode. This was due to the fact that (for the TEM mode) there are no z-components to the fields and there is no transverse curl in the electric field, so the integrals used to define the unique voltage and current along the cable were independent of integration path. However, at the end of the line, where the load is attached, this will no longer be the case. The EM fields surrounding the load will certainly not be perfect TEM fields. Therefore, the exact value of ZL will be ambiguous. For example, the voltage value V (z l) used in defining ZL will not be independent of integration path between points on the inner and outer conductors of the coaxial cable in the cross-section at z l , where the load is connected to the cable. A further problem also exists; it will be shown that the amplitude of the reflected power wave depends on ZL (and therefore the power transferred down the line depends on ZL). However, if the frequency is low enough, the phase of the EM fields will not change significantly in the region near the load, and the values for V ( z l ) and I ( z l ) will be nearly path independent [12]. If one can identify a specific path across the load at z l 39 along which the voltage and current integrals in Equation 2.35 can be computed, these values can be used to calculate an approximate ZL that can be used in further analysis. For the experiments discussed in this work, the network is always being operated within the low-frequency range. For a simple network, where a source is delivering power to a load via transmission line, generally the goal is to eliminate any reflected wave travelling back to the source. It will be shown that the incident wave and reflected wave carry power with them independently, so a reflected wave represents power that is not being transferred to the load. To eliminate the reflected wave, the transmission line must be impedance matched with the load. This means the effective impedance of the load must be equal to the characteristic impedance of the line. To show this, one begins by combining Equations 2.50 and 2.51 to give Vcoax ( z ) V0coax e I coax ( z ) V0coax Z 0coax i Z coaxYcoax z e V0coax e i Z coaxYcoax z i Z coaxYcoax z V0coax Z 0coax e i Z coaxYcoax z 2.53 . The reflection coefficient can then be defined as ( z ) V0coax e i Z coaxYcoax z i Z coaxYcoax z 0coax V e V0coax 0coax V e i 2 Z coaxYcoax z . 2.54 Hence, the reflection coefficient at a cross-section z is the ratio of the reflected wave voltage to the incident wave voltage at that particular cross-section. As we desire no reflected wave, the goal is to ensure a reflection coefficient equal to zero everywhere on the line. One can express the impedance of the line as a function of the reflection coefficient: 40 Vcoax ( z ) V0coax e Z coax ( z ) I coax ( z ) V0coax i e Z 0coax i ZcoaxYcoax z ZcoaxYcoax z V0coax e 0coax V i ZcoaxYcoax z e Z 0coax Z 0coax i ZcoaxYcoax z 1 ( z ) . 1 ( z ) 2.55 Rearranging to solve for the reflection coefficient in terms of impedance gives ( z ) Z coax ( z ) Z 0coax Z 0coax Z coax ( z ) . 2.56 This expression is subject to the end boundary condition (l ) V0coax V0coax e i 2 Z coaxYcoax l Z coax (l ) Z 0coax Z 0coax Z coax (l ) . 2.57 One can now solve for the voltage phasor ratio to obtain V0coax 0coax V Z coax (l ) Z 0coax Z 0coax Z coax (l ) e i 2 Z coaxYcoax l 2.58 . The reflection coefficient (Equation 2.54) may then be expressed as ( z ) Z coax (l ) Z 0coax Z 0coax Z coax (l ) e i 2 Z coaxYcoax ( l z ) . 2.59 It is now clear that in order for the reflection coefficient to be zero everywhere (no reflected wave), the impedance at the end of the line Z coax ( z l ) must be equal to (match) the characteristic impedance Z 0coax of the coaxial cable. 41 So far, we have examined how power travels down the coaxial cable, but only when a wave traveling in one direction is present (Equation 2.45). When both incident and reflected waves are present, the complex power, like the impedance, gains an imaginary part that will be a function of the z-position. However, the real part, which represents time-average power flow down the line, remains independent of position (as it must). To see this, one first expresses the voltage and current on the line in terms of the reflection coefficient at the end of the line ( z l ) : Vcoax ( z ) V0coax e I coax ( z ) V0coax Z 0coax i Z coaxYcoax z e (l )V0coax e i Z coaxYcoax z (l ) i 2 Z coaxYcoax l i Z coaxYcoax z V0coax Z 0coax e e i 2 Z coaxYcoax l i Z coaxYcoax z e 2.60 . Inserting these expressions into the definition of complex power gives Pf ( z ) V ( z ) I * ( z ) (V0coax e V0coax * ( Z 0coax i Z coaxYcoax z V0coax (l )V0coax e i Z coaxYcoax (2 l z ) 2.61 * e i Z coaxYcoax z (l ) Z 0coax e i Z coaxYcoax (2 l z ) ) ). The real part of Pf ( z ) is therefore given by Re( Pf ) V0coax Z 0coax 2 (l ) 2 V0coax Z 0coax 2 . 2.62 Hence, the total time-average power flow down the line is just the difference between the power that is carried independently by the incident and reflected waves. The imaginary part of Pf ( z ) is 42 Im( Pf ) 2 (l ) V0coax Z 0coax 2 sin[2 ZcoaxYcoax (l z ) ], 2.63 where we have explicitly shown the phase constant associated with the reflection coefficient. The imaginary part represents stored energy along the line; when there are both incident and reflected waves present a partial standing wave forms on the line which can store energy. This fact, and the way in which this stored energy varies with z-position, is important in many transmission line applications. For our case, since we desire the reflection coefficient to equal zero, the imaginary part will also equal zero when the line is impedance matched and will not be crucial to our analysis. 2.2.3 Inductive Coupling To this point, the characteristics of the EM field eigenmodes that can exist inside a RCCC have been determined, and the general concepts of power flow to a RCCC through a coaxial cable have been discussed. However, we have not discussed the exact manner in which these modes can be excited (or driven) by a power source. In short, some sort of current source must be present inside the cavity to stimulate the modes. Due to some dissipation of energy in a real cavity, this current source must remain inside the cavity, continuously replacing dissipated energy and keeping the modes running at a constant amplitude. Different current sources will stimulate different modes, and when a source stimulates a particular mode we say the source has “coupled” to that mode and is driving it. In this section, it will be shown how a properly positioned small loop of current can couple to the TM010 mode inside a cavity. Due to the nature of this particular coupling process, it is known as inductive coupling. 43 Thus far, it has been shown how (in general) power can be delivered to a load via a transmission line, specifically using the TEM mode of a coaxial cable. For our particular purposes, the load that will be receiving power from the transmission line is the combination of the RCCC and the small loop of current inside it. As we have also shown, this load (cavity and loop) must appear to the transmission line to be closely matched to the characteristic impedance of the line, otherwise power will be reflected before entering the cavity and no mode excitation will be significantly induced. In other words, the cavity and loop together must have an impedance equal to the 50 ohm impedance of the coaxial transmission line. The details of how this can be accomplished will also be discussed in this section. The basic arrangement for inductive coupling of a coaxial cable to a RCCC is shown in Figure 2-7. An aperture is created in the cavity wall that is roughly the same size as the outer conductor of the cable. The outer conductor is electrically connected flush with the surface of the aperture (and therefore the cavity boundary). The inner conductor of the coaxial cable is extended inside the cavity and curled into a loop, and is then electrically connected to the boundary of the cavity. If there is dielectric material inside the cavity, a small portion must be removed to allow space for the loop. Now recall that in order for the coaxial cable to operate in the TEM mode, boundary conditions on the inner and outer conductor surfaces require supporting sinusoidal surface currents to be present. When a TEM wave travels down the line and reaches the aperture surface of the cavity, as long as the wave is not reflected, the current on the inner conductor surface current will continue inside the cavity and around the loop. In this way, a source current is created inside the cavity which can couple to the different cavity modes. It is important to remember that the TEM wave will be reflected if the loop-cavity system is not properly impedance matched to the cable, but first it will be assumed that matching has been achieved, and there is now an 44 oscillating current source inside the cavity following the path of the loop shown in Figure 2-7. Figure 2-7 Inductive Coupling Illustration of inductive coupling using the central wire of a coaxial cable to form a small loop inside a cylindrical resonant cavity. We now examine the details of how the current source in the cavity will stimulate the cavity field eigenmodes that were defined in Section 2.1.3. First, the general case of an arbitrary current source coupling to modes of an arbitrary cavity will be considered. It will be assumed that a set of cavity eigenmodes have been obtained that satisfy Maxwell’s equations for the lossless (arbitrary) cavity with no current sources, Equations 2.3, with perfect conductor boundary Conditions 2.9. (This was accomplished for the specific case of the RCCC, with the TM and TE modes making up the eigenmode set.) The general eigenmodes will be labelled with index i, so that for the ith eigenmode the associated electric field will be written Ei, the magnetic field Hi, and the eigenfrequency i . As Equations 2.3 are satisfied by these modes, we have: (b) H i i i Ei . (a) Ei ii H i 45 2.64 Taking the curl of these equations then give separate wave equations for both Ei and H i : (a) Ei i 2 Ei 0 (b) H i i 2 H i 0. 2.65 Combined with the boundary equations, these two wave equations form an eigenvalue problem. For the lossless cavity case, the eigenvalues will be real, and the eigenfunctions Ei , H i form a complete orthogonal set [13]. This means that any arbitrary electric or magnetic field inside the cavity, even when losses and current sources are present, can be expanded in terms of the lossless eigenfunctions as follows, (a ) E Ai Ei (b) H Bi H i , i 2.66 i where the Ai and Bi are coefficients (phasors) for the ith eigenmode and must be determined. First it is helpful to normalize the eigenfunctions, which we choose to do in the following way: 0 i j , 1 i j Ei E j dV H i H j dV 2.67 where V is the volume of the cavity. One can choose the Ei to be purely real quantities, which will make the H i purely imaginary, by Equations 2.64. Next, internal cavity current sources and dielectric losses are introduced. Maxwell’s equations then become 46 (a) E i H (b) H i E E J s , and the corresponding wave equation for E becomes E 2 E i 2 E i J s . Inserting the eigenfunction expansion for the electric field A E i E 2.69 gives 2 Ei i 2 Ei i J s . i i 2.68 2.70 Using Ei i 2 Ei (Equation 2.65), we obtain (after simplifying and some rearranging): A ( i i i 2 2 ) i 2 Ei i J s . 2.71 One can now take the dot product of each side with the complex conjugate eigenfunction E j * and integrate over the volume of the cavity. Due to the orthonormal Conditions 2.67, every term in the sum except for i j equals zero, leaving Ai i i 2 2 i 2 47 J s Ei*dV . 2.72 This shows that if the current source inside the cavity is known, the coefficients for each coupled eigenmode can be computed. The total electric field in the cavity then becomes: E Ai Ei i i i Ei i i 2 2 2 J s Ei*dV . 2.73 The total magnetic field can also be determined using E i H (Equation 2.68a) and Ei ii H i (Equation 2.64a): H i ii H i i 2 2 i 2 J s Ei*dV , 2.74 and it is clear that the coefficients Bi from Equation 2.66(b) become: Bi ii 2 2 2 i i J s Ei dV . 2.75 It has now been demonstrated that if the current source inside a cavity is known (such as a small current loop positioned as in Figure 2-7, or a beam of electrons passing through the cavity (see Section 3.3.1), one can theoretically calculate the integral J s Ei dV which will give the amplitude of Ei and H i for every ith mode stimulated inside the cavity, in terms of known dielectric constants and eigenfrequencies. 48 In the above analysis, some key approximations have been made that need to be further examined. First, the boundary conditions that were implicitly used required the cavity to be completely enclosed by a perfect conductor. In application, the conducting boundary will not be perfect but will have a finite conductivity that will lead to power loss in the cavity walls as described in Section 2.1.4. This power loss will lessen the field amplitudes. To account for this, a parameter called the unloaded quality factor Q0 must be introduced. This parameter is related to the total loss in the cavity due to dielectric loss and the conduction losses in the walls. This parameter can be measured experimentally, as will be shown in Section 3.3.4, and the actual field amplitudes can be calculated using the equations of this section if the quantity (which accounts only for the dielectric loss) is replaced by 1/Q0 [11]. For example, Equation 2.74 would become H i ii H i i 2 2 iQ 2 J s Ei*dV . 2.76 0 Second, the cavity is not completely enclosed; there is an aperture where the coaxial cable meets up with the cavity. Furthermore, for our specific experimental application, there will also be two holes (apertures) in the ends of the cavity to allow the electron beam to pass through. These apertures come with a boundary condition requiring the transverse field components to be continuous across them (this condition can be quickly derived from Maxwell’s equations in the aperture). This requirement will distort the field. In addition, a current source was assumed to be present in the cavity, but we did not specify that this source was traveling through a metal conducting wire, as it will be in the case of the metal coupling loop. The conducting wire adds 49 new boundary conditions; the transverse component of the electric field must be zero on the surface of the wire inside the cavity. These new boundary conditions will also distort the field. Generally, the field distortions come in the form of higher order cavity modes being stimulated in the cavity, and these modes interfere destructively in all regions except in the near vicinity of the aperture(s) and the current carrying wire. In the perturbed regions, the higher order modes interfere constructively and arrange themselves to meet the actual boundary conditions. The region of constructive interference of the higher order modes dies off rapidly as you move away from these regions, and the field begins to resemble the solutions to Equations 2.73 and 2.74. 2.2.4 Impedance Matching of an Inductively Coupled Cavity A transmission line analogy has been used to describe the system of a coaxial cable delivering power to a resonant cavity. According to this analogy, the transmission line will be impedance matched when it “sees” the cavity as a load with impedance equal to the characteristic impedance of the line, in our case 50 ohms. In addition, when the line is impedance matched there will be no reflected power and all power from the source will be delivered to the cavity as desired. It is important to understand the limitations of this analogy, and why it can only be used for approximation purposes when attempting to design an actual coupling loop to be inserted in the cavity. To do this, one needs to analyze the EM field that exists in the plane of the aperture that connects the end of the coaxial cable to the cavity. Since there are no surface charges or currents in the aperture, boundary conditions require that the transverse EM field components be continuous across the aperture. This means that one cannot simply have, on the coaxial cable side of the aperture, a perfect coaxial TEM wave pattern with no reflected component, and then immediately across the aperture have a perfect 50 cavity eigenmode field. The TEM pattern in the cable and the TM 010 cavity pattern, for example, do not match up. Again, higher order modes must appear in the aperture in order to ensure that the fields are continuous across it. Higher order coaxial cable mode patterns have not been previously discussed, but they do exist, and together with the coaxial TEM modes form a complete orthogonal set of functions for the cable cross-section. This means that if the exact transverse electric field in the aperture were known, it could be expressed as a series expansion over the transverse coaxial cable eigenmodes as follows: ETrans E Trans z ETEM dS E Trans z ETEM dS ETrans Ei dS , 2.77 i where ETrans represents the actual transverse electric field in the aperture, and the integrations are over the aperture surface. The first two integrals represent the projected amplitudes of the TEM modes traveling in the positive and negative zdirections. The last term represents the projected amplitudes onto the higher order modes, which will not propagate down the cable but instead will die off exponentially away from the aperture. In order for these higher order modes to be sustained, energy will need to be stored in the region around the aperture. As far as the transmission line analogy is concerned, it is important to understand that a coaxial cable can only be considered a true transmission line when only the TEM modes are present. This is because it is the properties of the TEM modes that allow us to uniquely define a line voltage at any point along the cable to use in the transmission line equations. The existence of higher order modes in the cable near the aperture means the transmission line analogy is not technically accurate at the aperture. Unfortunately, the aperture region is critical in determining the reflection coefficient. 51 It is, at this point, informative to momentarily abandon our transmission line analogy and determine what is necessary, from a field standpoint, for the system to be impedance matched. An effective impedance of 50 ohms (real) means the reflection coefficient in the aperture is exactly zero. Certainly this requires the second integral in Equation 2.77 to be zero (no projection onto TEM mode traveling in the negative z-direction). It will also require the energy stored in the higher order modes to be equal parts electric and magnetic. One can see from Equation 2.29, derived by analyzing EM field power flow, that an imbalance between stored electric and magnetic energy is related to the imaginary part of complex power; but Equation 2.63 indicates that if an imaginary part of complex power exists, then the reflection coefficient cannot be exactly zero. It is now possible to appreciate the difficulty of solving the impedance problem exactly. Once a particular shape and size for the wire coupling loop has been designed, one would need to express the field on the cable side of the aperture as an infinite series in cable eigenmodes with unknown coefficients and, similarly, the field on the cavity side of the aperture would be expressed as an infinite series in cavity eigenmodes with unknown coefficients. All coefficients would then need to be solved for while satisfying the boundary conditions imposed by the aperture and the conducting metal wire. Once this is achieved all integrals in Equation 2.77 would need to be calculated. The second integral would need to evaluate to zero. The energy stored in all higher modes would need to be calculated and the total stored electric and magnetic energies would need to be exactly equal. If not, one would need to adjust our loop design and start over. Needless to say, this is more or less impossible. Many approximation methods of varying complexity have been developed [14]. These methods involve assuming a field in the aperture, calculating the induced surface currents on the wire loop to meet the field boundary conditions on the wire, and then applying Green’s theorem to the surface of the aperture, wire, and cavity to 52 determine the cavity modes stimulated (the Green’s function is written as a cavity eigenfunction expansion in the usual way). Once this is done, the fields that are calculated in the cavity can be added to the original assumed aperture field and an iterative process can begin, arriving at better and better approximations. Fortunately, these more advanced approximation methods are not necessary for our purposes. To obtain an estimate for the required size of our wire loop, the problem can be treated in a comparatively simple way by abandoning the transmission line definition of a load impedance for the loopcavity load. The transmission line definition would involve the ratio of a unique voltage (calculated in the aperture) to the current flowing down the central wire of the coaxial cable at the aperture. The presence of the higher order modes makes this definition unusable. Instead, we will define the loop-cavity load impedance as the ratio of the EMF around the wire loop, in the presence of the cavity fields, to the current flowing in the central wire at the aperture. In essence, the current flowing around the wire loop inside the cavity will stimulate fields in the cavity according to Equations 2.73 and 2.74. These fields will in turn create an EMF around the loop. This definition of the load impedance as the ratio of this EMF to the input current at the aperture was developed by Condon [15] and matches well with our loop design and experimental results. For inductive coupling to the TM010 mode, one is specifically interested in the magnetic field stimulated in the cavity near the loop current, because the periodic magnetic field passing through the loop is what will generate the EMF around the loop. Several assumptions will be made in this analysis, the first of which is that we are operating the system at the resonance frequency of the TM010 mode. This will cause the TM010 mode to be stimulated much more strongly than the other modes, which can then be neglected. From Equation 2.74 one can show that the H-field stimulated in the cavity will be: 53 H TM TM i010 H 010 i 010TM N 2 J E010 s TM * N dV TM Q0 H 010 TM 010 N * s TM J E010 N dV , 2.78 Q0 where we have replaced the quantity with the unloaded quality factor Q0 (since there will be loss in the cavity walls), and the subscript N refers to the normalized eigenfunctions. The next assumption is about the current in the loop. We assume the loop is small enough that the oscillating current is in phase around its entirety. It will also be assumed that, since the loop wire is thin, one can approximate the current density by simply dividing the total current by the crosssectional area of the wire. The volume element in the above integral will then be the cross-sectional area of the wire multiplied by a differential length of the wire. Equation 2.78 then becomes H TM Q0 H 010 TM 010 N Iin TM E010 Ac * N Ac ds TM Q0 H 010 TM 010 N I in TM E010 * N ds , 2.79 where we have defined Iin to be the current (phasor) in the central coaxial wire at the aperture (which we assume continues and remains constant around the loop), and Ac is the cross-sectional area of the wire. Next, it is assumed that the loop is a nearly closed circle so that Stokes’ theorem can be applied to the line integral, giving H TM Q0 H 010 TM 010 N Iin TM ( E010 54 * N ) dAloop , 2.80 where Aloop is the area vector for the loop. Equation 2.64(a) can now be used to replace the curl of the electric field, resulting in H TM Q0 H 010 TM 010 N I in TM TM i010 H 010 * N dAloop . 2.81 The integral now involves the dot product of the magnetic field and the loop area vector. Because of the way the loop is positioned in the cavity, its area vector will be pointing in the ̂ direction in our cylindrical coordinate system. The magnetic field for this mode is also pointing in the ̂ direction and its magnitude will remain relatively constant over the area of the loop, since the loop is small. The loop is near the outer radius of the cylinder, so the magnetic field magnitude, TM which is a function of the coordinate, will be approximately H 010 N ( R) over the area of the loop, where R is the radius of the cavity. The integral then disappears to give * TM TM H iQ0 I in H 010 ( R ) Aloop H 010 ˆ. N N 2.82 According to Faraday’s Law, this magnetic field will induce an EMF around the loop TM EMFloop i010 H loop Aloop , 2.83 where Hloop is the magnetic field through the loop. The magnetic field through the loop is just Equation 2.82 evaluated at R , implying that 55 TM TM 2 EMFloop Iin 2010 Q0 H 010 . ( R) Aloop 2 N 2.84 With the definition of load impedance, Zload EMFloop Iin , we finally obtain: TM TM 2 Zload 2010 Q0 H 010 . ( R) Aloop 2 N 2.85 This analysis clearly results in the impedance being real, primarily because we ignored non-resonant modes. The unloaded quality factor Q0 must eventually be measured experimentally but can be estimated in advance (values are typically 3000-6000 for these types of cavities), the normalized magnetic field can be evaluated using Equation 2.25 with normalization Conditions 2.67, and the required area of the loop can therefore be solved for with Zload 50 . As will be shown in the following chapter, this approximation is sufficient to allow a loop to be fabricated that achieves impedance matching (no reflected power). 56 3 Experimental Set-Up A diagram of the system used in the pulse compression experiments is shown in Figure 3-1. A brief description of the main components and how they are used together will be given here; more detailed descriptions and characterizations of certain individual components will be given in the remaining subsections of this chapter. Figure 3-1 System Components Major components for the synchronized laser-driven UEM column used in experiments. Experiments can be performed with or without the laser amplifier. The interferometer is used only in double pulse experiments. Not shown is the vacuum system as well as control electronics. The overall system is driven by a unique laser system that generates a pulsed laser beam. The pulsed laser beam in turn drives a photoelectron gun, which generates a pulsed electron beam. The electron pulses are accelerated by a large potential difference inside the gun, and then directed through the Compression Resonant Cavity (CRC) using horizontal and vertical deflector 57 plates. Magnetic lenses are used to control the transverse spread of the beam pulses, ensuring they pass through the CRC without contacting any of the inner surfaces. While inside the CRC, each electron pulse will interact with an EM field which is resonating in the TM010 mode, causing changes to the pulse shape and/or velocity. Depending on the phase of the EM field at the moment each pulse enters the cavity, the pulse can compress, expand, accelerate or decelerate. The goal of the system is to maximally compress the pulse while leaving its velocity unchanged. This means synchronization of the phase of the EM field with the arrival of each pulse is of the utmost importance. After passing through the CRC, and assuming the EM field has acted to compress the pulse, the pulse shape will continue to shorten until it reaches a maximum level of compression at the temporal focal point of the CRC. At this point the pulse passes through the Detection Resonant Cavity (DRC), which is capable of determining if the CRC EM field is at the proper phase for compression, and allows for proper adjustments to the optical delay stage. For long duration pulses (40-120 ps), the DRC can also measure the duration of the pulse and determine the power level when maximum compression has been achieved. A second set of magnetic lenses and deflector plates are used to guide the pulse through the DRC. The pulse then travels to a Faraday cup which monitors the electrons per pulse in the beam. As previously mentioned, synchronization of the phase of the CRC EM field with the arrival of each pulse at the CRC is critically important. To accomplish synchronization (and stability) of the EM field phase/electron pulse relationship, the same laser beam which drives the photoelectron gun is also used to drive the EM field in the CRC. A small amount of energy from the laser cavity is directed towards a fast photodiode with a sub-ns response (see Figure 3-1). The photodiode signal contains all the harmonics of the fundamental laser cavity frequency up to 10 GHz, and a particular harmonic matches the resonant frequency of the TM010 mode of the CRC. This harmonic is extracted from the 58 photodiode signal by a standard filter, and then sent to a 30 watt amplifier after pre-amplification by 20 dB. The high power signal is then sent via low-loss coaxial cable to drive the impedance matched CRC. Since the CRC and the photoelectron gun are being driven by the same source, there is no concern for phase drift of the EM field relative to the generation of the electron pulses at the photocathode of the 30 kV electron gun. The only concern is time jitter at the CRC, which will be shown to be low enough to allow successful operation of the system, due mainly to the high stability inherent to the mode-locked Yb:KGW laser system. The phase of the EM field relative to the electron pulse arrival is controlled by an adjustable optical delay stage which is part of the main laser beam optics. 3.1 Summary of Main Design Considerations Before describing the individual components of the system, a brief summary of the main design considerations (and how they affect the design of the individual components) for the system as a whole will be presented. The first consideration is the pulse charge requirement, or the number of electrons required in each individual electron pulse in the pulsed beam. The ultimate goal of the UEM project at UIC is to use a single electron pulse in a DTEM process to create an image of fast-moving phenomena (such as atomic vibration in a crystal). According to the Rose criterion [6], 100 electrons/pixel is required for acceptable gray-scaling of an image. For a typical 1k x 1k CCD camera with 106 pixels, this criterion equates to about 108 electrons per pulse [8]. Resolving a diffraction pattern requires about 106 electrons per pulse [7]. The large number of electrons per pulse, especially for imaging, creates the need for a highly controlled electron emission process in the electron gun, since a large percentage of the emitted electrons need to be directed through the system and to the sample. Typical emission processes used in traditional electron microscopy, such as thermionic 59 emission, can extract large amounts of electrons but in relatively random directions from the cathode in the gun; the small amount of electrons that happen to be emitted in the correct direction towards the sample are then used for imaging while the others are screened out. This type of process will not be adequate for imaging with DTEM however. Instead, a process is needed where the initial direction of the emitted electrons can be much more controlled, and less electrons are wasted. Therefore, a laser-driven photoelectric emission process must be used. This dictates the necessary power of the laser, the laser operating wavelength, the types of materials that can be used as a photocathode, the strength and shape of the gun acceleration field, as well as many other aspects of the system [16, 17]. The pulse charge requirement discussed above leads directly to another requirement on the system, which also affects the design criteria for multiple system components. There is a limit to the current density which can be extracted from a photocathode using a pulsed laser system. When the current density exceeds this limit, the electric field created by the just emitted electrons, still located a short distance from the cathode surface, shields the cathode from the acceleration electric field created by the gun voltage. Since the acceleration field is necessary for extraction of electrons from the cathode, useful extraction of electrons is greatly diminished beyond this limit. For long duration electron pulses, this limit is expressed by the Child-Langmuir law [18, 19]: J Pulse 4 0 9 32 2qe Vg . me d 2 3.1 Here JPulse is the magnitude of the current density of a single electron pulse, qe and me are the charge and mass of an electron, respectively, Vg is the acceleration voltage of the electron gun, and d is the spacing between the photocathode and 60 anode. This law needs to be modified when the duration of the emitted electron pulse is much less than the travel time of the pulse from cathode to anode inside the gun. The new limit becomes [20]: J Pulse 0Vg Dp d , 3.2 Where Dp is the electron pulse duration. When the electron pulse is short enough that this new limit must be used, the system is considered to be operating as an ultrafast or UEM system. This is not a distinction that is widely accepted in the literature but it is used by the UIC UEM group. Equation 3.2 enables the calculation of the minimum HW1/eM electron pulse parameter (wp) coming off of the cathode which will keep the current density below this short-pulse Child’s Law limit: wp ,min ne qe d , 0Vg 3.3 where ne is the desired number of electrons per pulse. For typical electron guns used in DTEM, Equation 3.3 gives a value for wp that is approximately 0.5 mm for extraction of 108 electrons. This is considered a relatively large transverse size for a pulsed beam, and the effects that this size requirement has on the design of system components is detailed in Chapter 6 of [21]. A common parameter used to characterize the overall quality of an axially symmetric Gaussian shaped pulsed electron beam is the normalized rms transverse emittance [22]: 61 T wp pT 2 mec 2 , 3.4 where c is the speed of light, and pT is the transverse momentum of an electron in the pulse. The pulsed beam is considered to be of higher quality if the transverse emittance is low, and it can be shown that in order to achieve a certain spatial resolving power, the transverse emittance may not exceed a certain value [17]. In Section 3.2 of [21], Liouville’s Theorem is used to show that the transverse emittance is conserved during electron pulse propagation. For perfect electron optics, it is unaffected by lensing, compression, etc.; the emittance of the beam can increase due to imperfect lensing [23], but it cannot decrease. Therefore, the restriction on the emittance value is present the moment the pulse is generated at the photocathode. We can combine this restriction with the minimum required value for wp at the photocathode discussed earlier, and arrive at a maximum allowed value for the initial standard deviation in transverse momentum for a pulse. Rearranging Equation 3.4, and using Equation 3.3, we find: pT 2 2 0Vg 2 T ,max mec T ,max mec. wp ,min ne qe d 3.5 Since the initial transverse momentum of the pulse is highly dependent on the exact process of photoemission, Equation 3.5 has led to extensive research and experimentation involving the photoemission process and the overall electron gun design used by the UEM group at UIC [24-28]. The design considerations discussed so far mainly correspond to the design and construction of the laser system and the photoemission process inside the electron gun. The last consideration involves the dynamics of a single electron 62 pulse from the moment after it leaves the photocathode to the moment it reaches the specimen to be imaged. After emission occurs, the pulse will experience a rapid acceleration inside the gun due to the high voltage between cathode and anode (the gun voltage); then upon leaving the gun the pulse will be focused both in the transverse and axial directions by magnetic lenses and the CRC, respectively. During this time the electrons in the pulse are of course repelling each other (the space-charge effect). A detailed understanding of the pulse dynamics is necessary to determine the proper design criteria for the magnetic lenses and the CRC. For example, a CRC can be designed to run at a wide range of frequencies and power levels. It can also be operated in any of the cavity eigenmodes. All of these characteristics will determine the exact focusing forces that a pulse will experience as it travels through the CRC. Knowledge of the pulse dynamics is therefore needed to determine the necessary forces for proper focusing in both space and time. A theoretical model of the electron pulse dynamics is utilized to determine the above mentioned design criteria. The model used by the UIC UEM group is based on the Analytic Gaussian (AG) theoretical model of Michalik and Sipe [29, 30], which includes a scheme to mimic the space-charge effects experienced by the pulse but excludes external forces. The model was extended by UIC UEM group member Joel Berger [31] to include external forces simulating gun acceleration, magnetic lensing and RF compression, and also to add initial conditions to the pulse propagation which simulate the photoemission process. Computer simulations using this extended model approximate the necessary lensing forces and proper positioning of various components. Knowledge of necessary forces allows for determination of certain design criteria for individual components, which will be described in the following sections. 63 3.2 Pre-existing Components As mentioned previously, the experimental results described in this document are part of a larger project conducted by the UEM group at UIC. Many of the components used in these experiments were previously developed and constructed by other members of the group and the details of their operation have been documented in various published papers and theses. Therefore, these components will only be discussed in this document in terms of their relevance to the current experiments, with references given so the interested reader can access any further detail desired about these components. 3.2.1 Laser Oscillator and Frequency Doubling In order to produce short electron pulses to be used for imaging, the electron gun in the UEM system is driven by an ultrafast laser system. The requirements of the photoemission process inside the gun, and the ensuing pulse dynamics, dictate the necessary characteristics of the short laser pulses hitting the cathode. The pulses must be below a certain duration, have the necessary photon energy (frequency) and the necessary pulse energy to generate the proper amount of electrons per pulse. The laser system used by the UIC UEM group was designed and built in-house by group members in order to meet the necessary pulse requirements. There are three main aspects to the system which will be briefly described in this section, they are; the crystal gain medium used in the oscillator, the thermal lens shaping technique employed to counter astigmatism inherent in the system, and the mode-locking process necessary for a pulsed output. Details regarding these aspects of the system (and many others) can be found in [16, 21, 32, 33] The laser oscillator uses a Ytterbium-doped potassium gadolinium tungstate (Yb:KGW) crystal gain medium. This material has a wide absorption spectrum around 980 nm, and a wide emission spectrum around 1040 nm. Absorption and emission efficiency are dependent on many factors, including the 64 polarization and incidence angles of the radiation relative to the crystallographic axes of the material. By employing a particular crystal geometry, along with the application of a 193 nm thick SiO2 anti-reflection coating, both of these properties can be exploited. The crystal is pumped by two 35 W laser diodes at 980 nm (about 94% absorption efficiency), so that access to the wide emission spectrum at 1040 nm is achieved. The wide laser gain spectrum allows for generation of ultrashort mode-locked pulses. The oscillator system employs a thermal lens shaping (TLS) technique [32] to account for astigmatism inherent in the particular crystal geometry. Based on the known angles involved in the crystal geometry and the crystal material properties, a mathematical analysis can be performed to determine the heat distribution required in the crystal to create the proper elliptical temperature distribution that generates the thermal lens compensating for astigmatism. This leads to specific requirements for design of the laser diode pumping system (the heat source), including the necessary elliptical pump spot size. The oscillator is capable of self-starting mode-locked operation (short pulse generation) above a certain power level. Mode-locking occurs in a laser cavity when a wide bandwidth of radiation is emitted, and the various cavity resonant frequencies of radiation present all oscillate with a constant phase relationship that creates constructive interference in a small temporal region (the pulse), and destructive interference everywhere else. The pulse then travels through the cavity with a particular group velocity and a pulsed beam becomes the output of the laser cavity, with a pulse repetition frequency equal to the fundamental oscillation frequency of the laser cavity. The output from the main oscillator cavity is a 28.5 MHz train of pulses with an average power of 1.9 W, a spectrum centered at 1046 nm, and a pulse duration of 250 fs. The photon energy at 1046 nm is generally not high enough to emit electrons from most photoemission sources [34]. To increase the photon 65 energy to a level appropriate for photoemission, the radiation from the oscillator undergoes two frequency doubling processes. The first process uses a 3 mm noncritically phase matched Lithium triborate (LBO) crystal [35]. The frequency doubling occurs with an efficiency of 40-50%, resulting in an output of 532 nm (green) radiation at a power of 0.8-1 W and leaving the pulse duration relatively unaffected. The second process employs a 6 mm -Barium borate (BBO) crystal which generates 261 nm (UV) radiation. This time the pulse-duration is significantly affected, due to group velocity mismatch effects, resulting in a pulse duration of approximately 4 ps. However, the photon energy is now 4.75 eV which is appropriate for photoemission from many photocathode materials. It is this 28.5 MHz UV radiation which strikes the photocathode in our experiments that do not involve the laser amplifier. 3.2.2 Laser Amplifier To produce the necessary pulse energy in the UV for the generation of sufficient electrons for single-shot imaging applications, the ultrashort pulse laser system dedicated to the UEM project at UIC includes a diode-pumped Yb:KGW regenerative amplifier. The design of the regenerative amplifier cavity is based on the same thermal lens shaping technology employed for the mode-locked Yb:KGW oscillator. In this case, the cavity is a simple z-fold around the gain medium with a Rubidium titanyl phosphate (RTP) transverse Pockels cell and sapphire Rochon polarizer [36] in one arm to facilitate the switching in and out of the amplified pulse. Standard chirped-pulse amplification techniques [37] are employed to avoid optical nonlinearities in the amplification process: 1000 lines/mm transmission gratings with approximately 90% efficiency in first order (Littrow configuration) both stretch the incident 250 fs pulse to nearly 100 ps prior to amplification and recompress the amplified pulse. Output amplified pulse energies of up to 0.1 mJ and compressed pulse durations of approximately 500 fs are obtained at operational repetition rates of around 600 Hz. These 66 compressed pulses are frequency doubled in a 2 mm BBO crystal and subsequently doubled again to 261 nm in the same BBO crystal employed to frequency up convert the Yb:KGW oscillator output; again yielding roughly 4 ps UV pulses, but with an energy of several micro-joules. 3.2.3 UEM Column Design All of the system components displayed in the assembly in the lower right corner of Figure 3-1 are considered to be part of the UEM Column (the Faraday cup is not a permanent column component but is used to terminate the column in the experiments). Since the electron beam must be contained in a high-vacuum environment to allow free passage of the electrons, the column is essentially a vacuum chamber constructed from purchased vacuum hardware, with working components contained inside the chamber. The hardware includes ports for use with various electrical pass-throughs to power internal components. Besides the vacuum hardware and high-vacuum system, the main pre-existing column components are the photoelectron gun, the first set of deflector plates, and the magnetic lenses. The photoelectron gun design is based on Togawa [38] and modified by Berger [17] to meet the needs of the UIC UEM group. The major consideration specific to the UIC group is the fact that the gun must accommodate a larger beam spot size on the photocathode, due to reasons described in Section 3.1. The gun consists of a Wehnelt cylinder with the photocathode positioned in the aperture, and a large aperture anode. Along with a set of deflector plates which help guide the electron beam down the column [21], the gun is enclosed in what is called the main UEM vacuum chamber. A valve in the bottom of this chamber opens the entire column to the vacuum pump system. The Wehnelt/photocathode assembly is mounted to the port aligner, which allows for precise alignment of the cathode. Positioned behind the port aligner is the high voltage pass-through, which provides the high voltage electrical connection to the cathode and creates the 67 acceleration field (gun voltage) between cathode and anode. The anode and the deflector plates are mounted inside the main chamber using special mounting hardware, which is also used to position a second set of deflector plates to be described later. The set of two magnetic lenses was designed and built in-house by Berger and the details of construction and operation can be found in [21]. The design was guided by the AG model of the pulse dynamics through the column. Again, a major design concern was the fact that a larger beam cross-section requires larger apertures in the lenses. Even though the CRC had not been built at the time the magnetic lenses were designed, the AG model could determine the beam divergence that would be caused by the CRC and show that the magnetic lenses could compensate for the CRC lensing in the pulse compression regime. Divergence in the beam is also caused by the gun anode aperture [21]; again the AG model could predict the resulting effect and this was also built into the magnetic lens design. 3.3 Compression Resonant Cavity (CRC) The Compression Resonant Cavity (CRC) is the first of two RF cavities to be built and installed in the UIC UEM system. The details of the operation, construction, and characterization of the device is described in the following subsections. 3.3.1 Operation During the photoemission process in the electron gun, each electron pulse will develop a certain level of temporal expansion due to velocity-induced dispersion; individual electrons will be emitted with a range of initial velocities due to the nature of the photoemission process, and electrons with higher velocities will move to the front of the pulse while slower electrons lag in the rear, as the pulse as a whole travels down the UEM column. Space-charge effects 68 can also contribute significantly to the longitudinal expansion of the electron pulse [7]. The purpose of the CRC is to reverse the resultant velocity chirp in each pulse, essentially slowing the faster moving electrons in the front and speeding up the slower electrons in the rear. After the pulse leaves the CRC, the electrons in the pulse will then begin to come together as the pulse moves through a section of drift tube, eventually converging to a minimum time duration at the temporal focal point of the CRC. A simple model can be employed to demonstrate the basic operating principle of the CRC and how it achieves velocity chirp reversal (and therefore pulse compression). In our experiments, the cavity is operated in the TM010 mode. The electric field of this mode has an overall sinusoidal time dependence, points in the z-direction, and has no variation in the z-direction at any particular time. The maximum value for the field occurs on axis where an electron pulse will travel. Therefore, the electric field interacting with the pulse as it travels through the CRC can be written as: Ez (t ) E010TM max sin(t ). 3.6 Here, the variable t is assumed to be zero as the pulse enters the cavity, and φ is therefore the phase of the electric field as the pulse enters the cavity. If the angular frequency of the field is written in terms of the cavity length, then the following analysis can be used to determine the ideal field phase constant and the ideal cavity length to achieve maximum compression. The force that any particular electron within the pulse experiences, due to this electric field, is 69 TM Fz qe E010 max sin(t ) 3.7 2 2 2 v , T Ad v Ad where d is the cavity length, v is the average electron velocity, and A is a unitless parameter (to be determined) which relates the period of the field to the cavity length. Since the field in the cavity is changing in time, as the pulse passes through any particular point along the CRC axis, the electrons at the front of the pulse will feel a different force at that point than the electrons at the back, which arrive at that point a pulse duration of time later. As the pulse duration is short (~4 ps) compared to the field period (~330 ps), it can be assumed the change in the field strength is linear, at any point, over the pulse duration. As a result, a simple expression for the differential force, between front and back, that the pulse (of duration τ) experiences at any point in the cavity may be written: Fz ,diff qe d TM TM E010 sin(t ) qe E010 max cos(t ). 3.8 max dt This expression can be integrated over the time the pulse is inside the cavity, to give the total compression impulse delivered to the pulse. This impulse is what causes reversal of the velocity chirp. To perform the integration we change variables from time to the cavity z-coordinate and then integrate over the length of the cavity: 70 I z ,comp qe E010 TM t t d cos(t )dt TM qe E010 max t 0 z d max z dz cos( v ) v z 0 d z qe E010 sin( ) max v 0 TM 3.9 2 v d sin( ) sin( ) max Ad v TM qe E010 max sin( 2A ) sin( ) . TM qe E010 For maximum compression impulse (which must be positive for compression), the quantity in square brackets must equal its maximum possible value of -2, which is achieved with A=2 and φ=π/2. The value of 2 for the parameter A means the cavity length should be set to allow for a pulse travel time through the cavity equal to one-half of the field period. The pulse dynamics of the AG model can predict the value for velocity chirp, and therefore the necessary compression impulse to reverse the chirp, allowing a calculation of the necessary field strength. It should be pointed out that, in addition to compression (or expansion), the CRC has the ability to accelerate (or decelerate) the center of mass of the pulse. To determine this acceleration one simply has to integrate the force experienced by the center of mass of the pulse over the time in the cavity, instead of the differential force experienced by the whole pulse. It is found that the phase of the EM field is what determines the effect of the CRC on the pulse. If the pulse arrives at the CRC when the phase is exactly π/2, the center of mass will not be accelerated and only compression occurs. Before attempting to determine the design parameters (dimensions, operating frequency, etc.) for the CRC, the effect of the cavity’s 2 mm axial hole must be investigated. Up to this point, the cavity eigenmodes that have been 71 studied were the solutions to a RCCC geometry that was assumed to be completely filled with a single dielectric material (or vacuum). When an axial hole is added, the problem is separated into two regions; the hole and the dielectric surrounding it. A general form for the solution to the field in each region is applied, and then boundary conditions are enforced at the outer conducting wall as well as at the interface between the hole (vacuum) and dielectric. It will now be shown that a 2 mm axial hole does not significantly affect the field pattern or resonant frequency of the TM010 mode of the CRC. For the region of the axial hole, the general solution for the field is simply 2 f i 2 ft E0 J 0 ( zˆ), e c 3.10 which is Equation 2.25 expressed in terms of frequency. For the dielectric region, the general solution becomes: 2 f r 2 f r i 2 ft BE Y . AE0 J 0 e 0 0 c c 3.11 Here the Bessel function of the second kind is included. This function has a singularity at 0 , but since the dielectric region no longer contains the axis, this second solution to the second order differential equation in Equation 2.20 cannot be discarded as it was previously. A and B are constants to be determined by boundary conditions. The first boundary condition requires the tangential Efield to be zero at the outer conducting wall 72 ( R) . This results in: 2 f r R 2 f r R AJ 0 BY0 0. c c 3.12 Next, the tangential E-field must be continuous across the interface between dielectric and vacuum ( a) : 2 f r a 2 f r a 2 fa AJ 0 BY0 J0 . c c c 3.13 Finally, the tangential H-field (Equation 2.25) must also be continuous across the interface: 2 f r a 2 f r a 2 fa A r J1 B r Y1 J1 . c c c 3.14 The above three boundary condition equations will have a solution provided 2 f r R J0 c 2 f r R Y0 c 2 f r a J0 c 2 f r a r J1 c 2 f r a Y0 c 2 f r a r Y1 c 73 0 2 fa J0 0. c 2 fa J1 c 3.15 Solving this equation provides the modified resonant frequency value for a particular axial hole radius a, and this value can be inserted into the boundary condition equations to solve for constants A and B, giving the field solution. A Mathematica program was written to solve for the eigenfrequencies of the YAG-filled CRC with axial holes of varying radius, with the results shown in Figure 3-2. From the plot it is clear that for small axial holes there is little effect on the resonant frequency. An experiment was performed to test the results of this theory (using a different dielectric material); the results are presented in Chapter 4, showing good agreement. Figure 3-2 Axial Hole Effect on CRC Frequency The effect of the axial hole, drilled through the YAG-filled CRC (radius 13.25 mm) to allow passage of electrons, on the cavity’s resonant frequency. For holes of small radius the change in frequency is negligible. The actual CRC axial hole radius is 2 mm. The Mathematica program was also used to solve for the unknown coefficients A and B and subsequently the field pattern with the 2 mm axial hole present. These results (with E0 1) are shown in Figure 3-3. It can be seen that the overall field pattern is only slightly disturbed by the presence of the axial hole. In particular, inside the hole the field magnitude is essentially the same, meaning 74 the electric force felt by the electron pulses passing centrally through the hole will remain the same. For these reasons, when doing any analysis involving the CRC field, we can replace the more complicated exact solution for the field with the solution for a fully filled cavity with no hole, using an effective dielectric constant that corresponds to the same resonant frequency as the exact solution. The dielectric constant for YAG at 3 GHz is estimated to be about 10.4 [39], and the effective dielectric constant for the CRC will be discussed in Section 3.3.4. Figure 3-3 Axial Hole Effect on CRC E-Field The electric field pattern for the CRC TM010 mode with and without the 2 mm axial hole, showing that the overall field pattern is not greatly affected, and the field magnitude in the hole (where the electrons interact with the field) is nearly identical. 3.3.2 Design There are many design parameters and constraints related to construction of the CRC. Parameters are entered into the AG model to determine if the requirements for pulse compression are met, and then various relations are used to check if these parameters allow successful operation of the CRC based on the various constraints. A process of trial and error eventually led to a set of parameters that meets all of the constraints and achieves proper pulse compression. 75 The main constraints on the CRC design will be discussed first. There are two size constraints; the CRC radius must be small enough to allow it to be easily mounted to the UEM column, and the temporal focal length of the CRC must be short enough to fit the length of the column. There is a constraint involving the velocity of the accelerated pulse as it enters the CRC; a high velocity is desired to reduce the time duration of the pulse for a given compressed length (improving time resolution), however, according to pulse dynamics simulations, a higher pulse velocity will involve a larger velocity chirp for each pulse, which in turn requires a stronger CRC E-field to reverse the chirp. This leads to power considerations; a stronger E-field increases the average stored electromagnetic energy in the cavity which will require extra power to maintain due to losses in the cavity. High power sources at RF frequencies are extremely expensive, and they become more expensive as the operating frequency increases for a given power level; this creates a constraint on the operating frequency. The velocity of the pulse itself is constrained by the maximum available voltage creating the acceleration field in the photoelectron gun. At extremely high gun voltages there will be problems with stability and arcing. The design parameters and their values will now be given, and it will then be shown that these values allow the CRC to operate within the aforementioned constraints. The first parameter involves the dielectric constant and loss tangent of the dielectric material chosen to fill the cavity. When the CRC is being run in the TM010 mode, the main purpose of a dielectric insert (with an axial hole to allow free passage of the pulses) is to reduce the radius of the cavity to more easily fit in the UEM column. This is because for a given frequency, the radius is proportional to 1 r . Any size reduction achieved must be weighed against increased power losses in the dielectric, so a low loss tangent is also desired. A crystal material with low loss as well as a large dielectric constant of 76 approximately 10.4 at 3 GHz is yttrium-aluminum-garnet or YAG [39], and this was the material selected for the CRC. Additional parameters that need assigned values include the operating frequency of the CRC and the photoelectron gun acceleration voltage. A higher frequency allows for additional size reduction and lower power needs, as will be shown. A frequency of 3 GHz was chosen, as high stability power supplies at this frequency are available and (relatively) affordable. The gun voltage was set at 25 kV, which determines the pulse velocity. The length of the CRC is chosen to correspond to a time of flight inside the cavity equal to a half period of the CRC EM field. This allows the pulse to enter the cavity when the field is at a maximum in the direction of the pulse velocity, and leave the cavity when the field is at a maximum in the opposite direction, allowing for maximum compression. With these chosen parameter values, AG model simulations showed a maximum CRC (time-dependent) E-field strength of approximately 0.2 MV/m would be necessary to completely reverse the expected velocity chirp (from the photoemission process combined with a gun voltage of 25 kV) with a temporal focal distance of 28.8 cm, an acceptable focal distance for the UEM column. What remains to be shown is that the CRC cavity size and the necessary power level are also acceptable within the constraints. The cavity dimensions are straightforward to evaluate. From Equation 2.24 the radius of the cavity is given by R 2.405c . 2 f r 3.16 For an operating frequency of 3 GHz and a dielectric constant of 10.4, the resulting cavity radius is 11.9 mm. However, since the axial hole was expected 77 to lower the effective dielectric constant of the CRC, a slightly larger radius of 13.25 mm was chosen. Since the time of flight for a pulse inside the cavity should ideally be equal to a half-period, the cavity length (denoted by d) is simply related to the pulse velocity (vp) and the frequency, which can be further related to the gun voltage: d vp 2f 1 2f 2Vg qe me 2.963 105 Vg f . 3.17 For a gun voltage of 25 kV and a frequency of 3 GHz, this results in a CRC length of 15.6 mm. These dimensions (radius and length) for the CRC allow the cavity to easily fit in the UEM column. One can now show that the CRC will operate under the stated parameters at an acceptable power level. First, the unloaded quality factor Q0 must be defined for a resonant cavity in terms of the cavity’s stored energy and power loss. In general, a quality factor for any oscillating physical system is defined as the ratio of time-average stored energy, to time-average energy lost per radian of a cycle. There can be different quality factors defined for a system depending on which type of stored energy, and which type of energy loss, are being referred to in the ratio. The unloaded quality factor for the CRC refers to the total time-average stored EM energy in the CRC and the total time-average energy loss per radian: Q0 2 f (We Wm ) . Pf Pd , 3.18 Here energy and power terms are used that were defined previously in Equations 2.28 and 2.34. For the CRC, the power terms represent the time-average power loss in the conducting walls and the dielectric loss. The total time-average cavity 78 power loss PT can be defined as the sum of these two terms. Equation 3.18 can then be rearranged to give PT 2 f (We Wm ) 4 f We , Q0 Q0 3.19 where we have used the fact that stored electric and magnetic energies are equal at resonance. This expression now needs to be expressed in terms of the design parameters. First, we replace the stored energy term: 4 f 12 E010 ( ) PT Q0 TM 2 R 2 TM 2 4 f d E ( 0) J d 0 r 010 0 0 dV Q0 3.20 2 TM 2 4 2 f 0 r d E010 ( 0) (0.135 R ) Q0 . Next, Equations 3.16 and 3.17 are used to replace R and d in terms of the design parameters, and the electric field phasor magnitude is replaced with the timedependent field magnitude (which is the actual field design parameter): 2.405c Vg 1 TM 2 5 4 f 0 r 2.963 10 E010 ( 0) 0.135 f 2 2 f r PT Q0 2 2 3.21 9.195 1010 E010TM ( 0) 2 Q0 f 2 Vg . 79 With the values given for the design parameters, this equates to a power level of 12.9 W if the CRC unloaded quality factor is 5000. Factors of 5000 are easily attainable for resonant cavities, and the CRC in fact has an unloaded quality factor of 5080 as will be shown in Section 3.3.4. A 30 W power source at 3 GHz is available to the UEM lab, so the design constraints have been met. 3.3.3 Construction The components and subassemblies that make up the CRC are shown in Figure 3-4. First a general description of the construction and assembly of the major components will be given, and more detail will then be provided as needed. The cavity is located in the center of a steel housing piece which allows the CRC to be mounted to the UEM column. There is a copper insert in the center of the housing, and the inside surface of this insert makes up the cylindrical boundary of the cavity. The YAG dielectric insert fits into the cavity and is secured by two end caps which are screwed onto the main housing. The center circular surfaces of the end caps act as the end boundaries of the cavity. To deliver power to the cavity, a hole is drilled through the steel housing and copper insert to allow insertion of the coupling loop assembly. The loop assembly is attached to a vacuum pass-through with SMA (SubMiniature version A) coaxial connectors, and mounted to a vacuum flange at the end of a steel tubular extension that is welded to the housing. When mounted, the small coupling loop at the end of the loop assembly will enter the cavity and fit into a notch drilled into the side of the YAG insert. The vacuum pass-through has a female SMA connector on the nonvacuum side which allows connection to the power source. 80 Figure 3-4 CRC Components Individual components of the Compression Resonant Cavity. The end caps and steel housing with copper insert, along with the loop assembly, were designed and constructed in-house. Figure 3-5 shows detail of the coupling loop construction. A small length of coaxial cable is fitted with an SMA female connector at one end. The other end is stripped so that a small length of the central copper conducting wire is left exposed. A machined brass end piece is press-fit over the end of the cable, and the central wire is bent into a loop. The end of the wire is then press-fit into a hole drilled into the brass end piece, which is electrically connected to the cable’s outer conductor. As the loop assembly is being inserted into the cavity, the wider section of the brass end piece acts as a stopper so the loop is inserted the proper distance. Once the loop is inserted into the cavity and properly oriented, a set screw (shown in the figure on the right) is used to tighten the end piece in place. 81 Figure 3-5 Coupling Loop Detail Detail of the coupling loop used to deliver power to the CRC. The loop can be rotated while inside the CRC, and fixed in place using the set screw once impedance matching has been achieved. Figure 3-6 shows detail of the copper insert, the through-hole for the coupling loop, the through-hole for the set screw, and how the coupling loop fits into the cavity and is tightened by the set screw (shown without the YAG insert present, so the loop can be clearly seen). The copper insert is machined so its outer diameter is slightly larger than the diameter of the central hole in the steel housing. The housing is then heated, and a cooled copper insert is fitted into the hole in the housing. When the assembly reaches room temperature the copper is securely fit into the housing, as shown in the picture. The loop assembly throughhole is then drilled through both the steel and copper and into the cavity. Another small through-hole is drilled into the edge of the copper insert and tapped to allow for a set screw as shown. When the loop is inserted, the end surface of the brass end piece actually becomes part of the cavity boundary. The set screw can then be tightened to hold the loop in place (although during actual assembly this is done after the end caps have been attached). The inside surface of the copper is highly polished to decrease the effective skin depth of the copper, which minimizes conduction losses in the cavity walls. 82 Figure 3-6 Cavity/Coupling Loop Detail Top: steel housing, cavity, and coupling loop through-hole are shown. Bottom Right: close up of the highly polished inner copper surface, loop through-hole and the through-hole for the loop tightening set screw. Bottom Left: close up of the inserted coupling loop and set screw. Figure 3-7 shows the detail of the copper end caps and the inside of the cavity with the back end cap attached. In the top picture in the figure, the “back” end cap is shown on the right. The only difference between the two pieces is a through-hole (seen in the picture) drilled into the “front” end cap to allow tightening of the set screw. Both caps have a circular groove milled into their surfaces to allow for a strip of indium wire to be placed (the front cap is shown without the wire so the groove can be seen clearly). The surface of each cap that is inside the groove will act as a boundary for the cavity when attached, and these inner surfaces are highly polished. In the bottom right picture in the figure, the back end cap has been screwed onto the housing, and the close up picture on the 83 bottom left shows how the indium wire, which is extremely malleable, fills in any gaps between the end cap and the central copper insert. This is extremely important due to the current pattern of the TM010 mode shown in Figure 2-3. Any gap between end cap and cylindrical copper insert will cause major disruptions in the current flow, and this greatly affects the field pattern and resonant frequency (as well as impendence matching and the total quality factor). The indium wire effectively stabilizes the current flow and allows for consistent performance of the CRC. Figure 3-7 End Cap Detail Detail of the CRC end caps showing how the indium wire is used to aid in current flow around the end boundaries of the cavity. Figure 3-8 shows a close-up picture of the YAG dielectric insert on the left. The axial hole as well as the clearance notch for the coupling loop can both be clearly seen. The surface of the YAG insert is highly polished which allows for the possibility of a copper coating process, which was successfully attempted using a fused silica dielectric insert shown on the right side of the figure. This 84 process leaves the dielectric with about a 60 micron thick coating of copper, which is much greater than the copper skin depth at 3 GHz and therefore serves as an appropriate cavity boundary surface. The copper coating also allows for improved current flow in the TM010 mode due to the indium wire (and end caps) no longer being involved in the flow. While the indium allows for more stable current flow for an uncoated dielectric-filled cavity with end caps, it still causes minor disruptions in the flow pattern due to varying thickness and slight protrusions into the cavity when the end caps are tightened down. This creates a less efficient production of magnetic flux by the current surrounding the cavity, essentially reducing the effective inductance of the cavity. The effect is similar to reducing the inductance in an LC circuit, and it is well known that a resonant cavity can be modeled as an equivalent LC circuit for purposes such as calculating the resonant frequency [11]. In an LC circuit, a reduction in inductance will cause the resonant frequency to increase. This is indeed the case for the fused silica insert. The resonant frequency of a cavity consisting of an uncoated fused silica insert, end caps and indium wire was measured to be 6% higher than the coated fused silica cavity. The unloaded quality factor of the coated cavity was also 20% higher signaling less loss in the coated walls. These same results would be expected for a coated YAG insert. 85 Figure 3-8 YAG Dielectric Insert and Coated Fused Silica Left: YAG dielectric insert showing axial hole and clearance hole for the coupling loop. Right: Fused silica dielectric coated with a 60 micron thick layer of copper, forming a stand-alone resonant cavity. For implementing a coated cavity in compression experiments, the mounting process would still be the same, and the same housing hardware can be used. The end caps and copper insert would simply be holding the coated insert in place, and would no longer be an actual part of the cavity. The coated fused silica cavity is shown inserted into the DRC housing in Figure 3-9. Figure 3-9 Coated Fused Silica Cavity in DRC Copper coated fused silica cavity inserted into the DRC housing to allow for testing and characterization. 86 For CRC experiments, the YAG insert was left uncoated and the indium wire was used to create stable current flow. This was due to the fact that an RF filter had already been selected that matched the uncoated cavity’s higher frequency. While a 20% increase in unloaded quality factor would be desired and a coated cavity is the better option, it was not necessary to successfully perform experiments. Figure 3-10 shows the next step of CRC assembly, the insertion of the YAG dielectric and the coupling loop. In the bottom right photo, the dielectric has been fit into the cavity with the notch for the coupling loop aligned with the loop through-hole. In the bottom left, the loop assembly has been inserted (the loop can be seen under close inspection) and the set screw is being tightened. The top photo shows the entire housing with YAG and loop assembly inserted. 87 Figure 3-10 YAG Insert with Coupling Loop CRC housing with YAG insert in place and coupling loop assembly inserted into the housing. In the lower left picture the loop can be seen inside the YAG clearance hole (upon close inspection), ready to be fixed in place by the set screw. Figure 3-11 shows detail of the loop assembly and vacuum pass-through being inserted into the housing. On the left the loop assembly is shown connected via SMA connectors to the purchased vacuum pass-through component. The top right photo shows the top of the tubular extension with a purchased vacuum flange welded on. The extension piece itself is welded onto a flattened section of the main steel housing as can be seen in the photo. In the bottom right photo a copper gasket has been placed into the groove of the vacuum flange. Next, in the middle top photo, the loop assembly is inserted into the cavity until the brass stopper prevents any further insertion. Finally, the middle bottom photo shows the top vacuum flange screwed down creating the vacuum seal. As the top flange is tightened down, the copper gasket will flatten a small amount and push the loop assembly further into the housing. Since the brass stopper prevents the loop from 88 moving further into the cavity, the coaxial cable will be forced to bow to the side slightly to account for this extra insertion. Thus, a minimum length of cable must be used in the loop assembly. Figure 3-11 Vacuum Pass-Through Assembly Left: vacuum pass-through connected to the coupling loop assembly via SMA connector. Right: various stages of insertion into CRC housing with vacuum seal. Once the top vacuum flange has been put into position, but before the screws have been tightened, the loop assembly can still be rotated, changing the orientation of the loop inside the YAG insert (the notch in the insert allows for 90 degree rotation). This is important for impedance matching purposes. During the actual impedance matching process the CRC will be hooked up to a network analyzer (after the front end cap has also been attached) to determine when the CRC is properly matched. The exact loop area necessary for impedance matching can only be estimated beforehand, as mentioned in Section 2.2.4. Therefore, in order to match the cavity, the loop is rotated slightly (changing the effective area of the loop receiving magnetic flux) until impedance matching is seen on the 89 analyzer. At this point the set screw is tightened (there is a through-hole in the front end cap allowing access to the set screw), fixing the loop in place. Lastly, the screws of the top vacuum flange are tightened down and the CRC is ready for insertion into the vacuum system. Figure 3-12 shows the fully assembled CRC. In the bottom picture the small through-hole in the front end cap is shown with a wrench tightening the set screw. The top photo shows how a copper gasket fits into a machined groove in the steel housing (an identical gasket is fitted to the other side). The CRC can now be mounted to the UEM column using the large threaded outer holes of the housing, which compress the gaskets and create a vacuum seal. Figure 3-12 Completed CRC Assembly Completed CRC assembly ready to be mounted to the UEM column using large copper vacuum gaskets. The end cap through hole allows the coupling loop to still be rotated for impedance matching. 90 3.3.4 Characterization A resonant cavity (operating in a particular mode) can be fully characterized by identifying three parameters: the resonant frequency, the coupling constant κ, and the total quality factor. The coupling constant is related to how well the cavity is impedance matched, and will be equal to unity if the cavity is perfectly matched, meaning the impedance of the cavity and coupling loop combined is 50 ohms at the resonant frequency. To identify these parameters, the frequency response of the coupled cavity in the vicinity of the resonant frequency must be analyzed. To perform this analysis, a resonant cavity is typically connected to a network analyzer, which sends out a swept frequency reference signal. The signal (magnitude and phase) that is reflected back to the analyzer by the cavity is then recorded at each frequency. A computer program is then used to analyze this data and obtain the parameters. An expression for the frequency response of a resonant cavity (operating in the TM010 mode) can be developed from the theory already discussed, in order to gain an understanding of how the computer program analyzes the data, but more importantly to show how the size of the coupling loop has a critical effect on impedance matching. First, a frequency-dependent expression must be developed for the load impedance of the CRC. In Section 2.2.4 a simplified version was provided to help approximate the necessary coupling loop size. There, the cavity was assumed to be operating at exactly the resonant frequency, which eliminated the frequency-dependent part of the expression. The frequency dependence simply needs to be put back in and the identical development can be repeated. Starting with Equation 2.76, repeated here specifically for the TM010 mode (the other modes can still be neglected since we are interested in the frequency range very near the TM010 resonance); 91 H 010 TM TM TM i010 H 010 TM 010 2 2 N iQ 2 s TM J E010 * N dV , 0 the volume integral can be evaluated the same way as before giving, TM H 010 TM 010 TM 010 2 2 2 I H TM * ( R) A H TM ˆ. loop 010 N in 010 N i Q0 2 3.22 After repeating the remainder of the derivation, an expression is obtained for the load impedance as a function of frequency near the TM010 resonance: TM i 010 2 Z load TM 010 2 2 2 H TM ( R ) 2 A2 . 010 N loop 2 i Q0 3.23 The real and imaginary parts of the complex quantity Zload are plotted in Figure 3-13. 92 Figure 3-13 CRC Impedance Real and imaginary components of Zload for a CRC with a 3 GHz TM010 resonant frequency, a loop area of 1.85×10-6 mm2, and Q0 of 5080. This frequency dependence of the CRC impedance is immediately recognized as having the same general form as the frequency dependence of a lumped element resonant RLC circuit. In fact, resonant cavities are often modeled as equivalent RLC circuits, and the computer program used to determine the cavity parameters from the network analyzer data is based on this type of modelling. Equation 3.23 can be re-written in the following form: TM 2 TM Q0010 H 010 N ( R) TM 010 iQ0 TM TM 010 010 2 Zload 2 Aloop . 3.24 The expression in the square brackets has units of inductance (due to the normalization conditions on the magnetic field eigenfunctions), and can be thought of as the inductance of the equivalent RLC circuit. Using the following relation from analysis of RLC resonant circuits (Q being the quality factor of the circuit), 93 R Qres L, TM and the fact that 010 3.25 1 near the resonance, Equation 3.24 can be simplified as follows: Z load R TM 010 1 iQ0 TM 010 3.26 . This is not the exact impedance as seen by the network analyzer, however: there will be a certain length of transmission line connecting the analyzer to the CRC, and a shorted transmission line (which is what the coaxial cable terminated in a loop becomes as soon as you move away from resonance) has a reactance that is a function of the length of the line as shown in Equation 2.55. This means a series reactance must be added to the expression for the load impedance to match what the network analyzer “sees”; Z load iX ( ) R TM 010 1 iQ0 TM 010 . 3.27 This expression matches the form of Equation (3) in reference [40], and this paper (along with [41]) details the process of determining the CRC parameters starting with Equation (3), using a network analyzer and the equivalent circuit model. To investigate the effects of coupling loop size on the CRC parameters, the reactance associated with the transmission line can be neglected and Equation 3.24 can be used. In Figure 3-14, the load impedance is plotted in the complex 94 plane, as the frequency is varied across the resonance, for three different loop sizes. Figure 3-14 Effect of Loop Area on Impedance Zload for a CRC with a 3 GHz TM010 resonant frequency, and Q0 of 5080, plotted in the complex plane as frequency is varied across the resonance. The typical circular pattern is shown for three different coupling loop sizes. As can be seen from the figure, each plot forms a near perfect circle. As we have neglected the transmission line reactance, the points corresponding to the resonant frequency fall on the real axis on the right-hand side of each circle. The strong sensitivity of the impedance to loop size is clear. As a result, the exact loop size needed for impedance matching can only be approximated beforehand, and the CRC construction must allow the loop to be rotated while it is connected to the network analyzer. As the loop is rotated the effective area and therefore the impedance varies rapidly, and careful tuning is required. Figure 3-14 shows that for the parameters entered into Equation 3.24, an effective loop area of 1.9×10-6 m2 will result in an impedance of approximately 50 ohms at the resonant frequency, which represents successful impedance matching. Again neglecting the effects of the transmission line, the load impedance can be transformed into the CRC reflection coefficient according to 95 load ( ) Zload ( ) 50 . Zload ( ) 50 3.28 The reflection coefficient is plotted in the complex plane (as frequency is varied across the resonance) in Figure 3-15 for the same three loop areas. Again the plots are near perfect circles, and again the effect of loop area is clear. Figure 3-15 Effect of Loop Area on Reflection Coefficient The reflection coefficient, for a CRC with a 3 GHz TM010 resonant frequency, and Q0 of 5080, plotted in the complex plane as frequency is varied across the resonance. The typical circular pattern is shown for three different coupling loop sizes. There is no new information contained in the reflection coefficient plots; they are shown here because the computer program, called QZERO, which determines the CRC parameters, uses the reflection coefficient data. The output from this program for the YAG-filled CRC is shown in Figure 3-16. The program plots the reflection coefficient in the complex plane for 201 different frequencies near the resonance. The output gives five parameters for the CRC, (only four are independent): QL, Q0, κ, κs, and fL. The loaded resonant frequency, fL, is defined 96 as the frequency corresponding to the smallest magnitude reflection coefficient. It is technically different from the unloaded resonant frequency, which is defined as the frequency the cavity would resonate at if the coupling loop were not present (since the loop has a small inductance of its own, it detunes the cavity slightly). The difference is so small that it can be neglected, and the value of fL will be considered the operating resonant frequency of the cavity. The QZERO program gives a value of 3.0597 GHz for fL. Figure 3-16 QZERO Output for CRC Output of the QZERO program for the CRC. This program takes as its input the reflection coefficient data from a network analyzer, and gives as its output five characterization parameters for the CRC, most important being the unloaded quality factor Q0. The coupling coefficient κ is defined as the ratio of power loss in the “cavity” to the power loss external to the “cavity”. In the circuit model, the “cavity” is the RLC circuit only and is considered the load impedance, and the rest of the set-up- the coupling loop, transmission line, and internal network analyzer impedance, is modelled as an “external” circuit. It is well known that in order to deliver maximum power to a load from a source, exactly half of the total power loss must occur at the load, and the other half occurs in the external circuit. 97 It has also been shown that maximum power is delivered to a resonant cavity when the line is perfectly impedance matched. This means that a coupling coefficient κ equal unity in the circuit model corresponds to perfect impedance matching. The QZERO program gives a value of 0.996 for κ. There is a second coupling coefficient included in the output of the program, κs, which is an attempt by the program to determine power loss occurring in the coupling loop alone. The value is masked by the transmission line losses in our case and can be neglected. Typically, losses in the coupling loop are much less than 1% of total loss and keeping track of this small loss is only important in the highest precision measurements, which is not necessary for the CRC design or operation. The loaded quality factor, QL, related to power loss in the entire circuit, is defined as QL Q0 (1 ) . The analyzer actually measures QL and κ directly and then uses this relation to arrive at Q0 for the cavity. The ability to accurately measure Q0 is the main advantage of the QZERO program (an acceptable approximation for fL and κ can be reached simply by viewing the network analyzer screen without need for additional computer analysis). The program is also able to approximate the error in its calculation for Q0 (as well as κ), which is included in the output. QZERO gives a value of 5084.8 ± 187.0 for Q0. The three parameters necessary for successful operation of the CRC have now been determined, but further characterization is possible since the CRC is filled with a dielectric material. The parameter Q0 provides information on the total power being dissipated in the cavity, which by itself allows for determination of the appropriate power source. However, it is also desirable to know how much of that power is being dissipated in the cavity walls versus the dielectric losses. There are many dielectric materials to choose from, and there are different ways of constructing the cavity walls, so this information is helpful for design purposes. Specifically, it is desired to compare loss in the walls for the CRC to loss in the 98 walls for a copper coated dielectric cavity. As mentioned earlier, it is found that the copper coating process reduces wall loss by 20%. The calculations are straight-forward. Equation 3.19 can be used to calculate the total power loss in the cavity (now that Q0 has been determined), and Equation 2.34 can be used to calculate the power loss in the copper walls only. The difference is considered the power loss in the dielectric (although there may be other small sources of loss, such as the coupling loop and indium wire). The results give a total power loss of 12.37 W for the CRC, with a 10.47 W loss in the copper walls and a 1.9 W loss in the dielectric. It should be pointed out that the above calculations used a value for the effective dielectric constant of the CRC, as calculated by Equation 3.16 using 3.0597 GHz for f and 13.25 mm for R. The value for the effective dielectric constant is 8.01, significantly different from the approximated value for the dielectric constant of YAG at 3 GHz, which is 10.4 (this value has never been directly measured at 3 GHz but is extrapolated from data at other frequencies). Some discrepancy is to be expected, since the CRC has the vacuum-filled axial hole as well as a notch to fit the coupling loop, so it is not completely filled with YAG to begin with. Also, as mentioned earlier, the indium wire and end cap construction increases the resonant frequency by about 6% compared to a copper coated cavity construction. This difference is also absorbed into the effective dielectric constant value despite being unrelated to the characteristics of the YAG material. An effective dielectric loss tangent can also be calculated for the CRC cavity. For a dielectric material, the loss tangent is defined as the ratio , which from Equation 2.28 can be seen to be equal to Pd , 2We using the actual value for r for the dielectric. To get the effective value for the CRC we again perform the same calculation but using the effective r . The result is a CRC loss 99 tangent of 3.03×10-5, which would be considered extremely low loss for a pure dielectric material. All of the parameters for the CRC which were determined by the network analyzer data only hold for operation under low power, since the reference signal of the analyzer is a low power signal. During experiments, the CRC will experience up to 20 W of RF power, which will cause significant heating. This heating will of course cause the cavity to expand, changing the dimensions and therefore also the resonant frequency. From the network analyzer data, it is clear that even the smallest change in the resonant frequency will have a large effect on the impedance matching. Since the coupling loop cannot be adjusted during operation, the impedance matching will continue to worsen as the CRC heats up and eventually power will be completely reflected by the cavity. To eliminate this problem, a cooling system was built and installed surrounding the cavity. Unfortunately, the network analyzer cannot be used to monitor the resonant frequency during operation in order to test the effectiveness of this cooling system, since the CRC must be connected instead to the high power source. However, the high power source does give a read-out of forward and reflected power from the CRC during operation. With the cooling system on, the power was slowly increased up to 20 W while the reflected power was monitored. The result showed that the reflected power varied slightly during operation, meaning the impedance matching was varying slightly, but the effect was small and so the CRC was considered sufficiently stabilized. The success of the high power compression experiment described in the next chapter is also evidence that the CRC is stable under high power operation. 3.4 Drift Tube and Deflection Plates The temporal focal distance for the CRC is approximately 30 cm, so there needs to be this amount of distance separating the CRC (where the pulse velocity chirp is reversed) and the DRC (where the pulse is focused). A drift tube is 100 installed in the UEM column to provide this distance. In order to help aim the beam through this extra distance and subsequently down the central axis of the DRC, a second set of deflection plates are included inside the drift tube. The tube is also wrapped in a layer of impermeable material, shielding the inside of the tube from the earth’s magnetic field, which will otherwise deflect the electron pulses off of the central axis at 25 keV. In Figure 3-17, the deflection plates are shown attached to a vacuum flange which will be bolted to the end of the drift tube. The electrical leads which allow the plates to be charges can be seen in the photo, and vacuum pass-throughs will be necessary so the leads can be connected to a voltage source. Figure 3-17 Deflector Plates Deflector plate assembly specially designed for insertion into the drift tube. Figure 3-18 shows the drift tube assembly. The tube itself is made of steel and has vacuum flanges welded to each end. The deflector plate assembly has been flipped over and inserted into the top of the drift tube. On the right, the detail of the electrical lead pass-throughs is shown (in these photos the impermeable cover for the tube is not shown). 101 Figure 3-18 Drift Tube Assembly Detail of the drift tube assembly with deflector plates and vacuum pass-throughs. On the right, the alignment of the pass-throughs and the electrical leads for the deflector plates is shown. 3.5 Detection Resonant Cavity (DRC) The construction and characterization processes for the DRC are basically identical to the CRC and do not need to be repeated (actually the DRC processes are simplified by the absence of a dielectric insert). The design parameters are also similar, with the differences related to the operation which is described in the following sections. Therefore, the QZERO parameters characterizing the DRC will simply be given here: fL=5.5408 GHz, κ=0.853, and Q0=9824.8±274.4. 3.5.1 Operation The DRC operates in a reverse manner compared to the CRC. Operation of the CRC is based on the coupling loop, carrying a single frequency current source, driving the cavity field which then acts on the electron pulses. In the DRC, the pulses themselves act as a current source and drive the resonant mode (at a much weaker power level) as they travel through the cavity, and this field creates an EMF around the coupling loop. The loop is connected via vacuum 102 pass-through and coaxial cable to an oscilloscope, where the loop EMF can be directly observed. The oscilloscope performs a FFT on the time-base data, and shows the voltage level around any driven resonant frequency of the cavity. This voltage is directly related to the duration of each pulse in the pulse train passing through the cavity, as will be shown below. Equation 2.74, repeated here for convenience, was used previously to determine the magnetic field inside the cavity when driven by current in the coupling loop: H i ii H i i 2 2 i 2 J s Ei*dV , 3.29 Q0 where Q0 here is related to the power loss in the cavity walls only, since the DRC is a vacuum-filled cavity. The same equation may be used to determine the field inside the cavity when it is driven by the pulses passing through it. From the derivation of Equation 3.29 it is evident that the current density source in this equation, J s , is a function of position and frequency. It is therefore the Fourier transform of the same source represented as a function of position and time. The source itself is a train of evenly spaced electron pulses traveling down the central axis of the cavity. As a first step, a mathematical expression for a single traveling pulse will be developed as a function of position and time. The Fourier transform of this expression will then be taken, and the result can be inserted into Equation 3.29. Since the transform of a traveling pulse contains a distribution over many frequencies, Equation 3.29 will need to be integrated over frequency to determine the total field: 103 H d i ii H i i 2 2 iQ 2 J s (r , ) Ei*dV . 3.30 0 The single pulse current source density is modelled as a continuous charge density distribution moving with a constant velocity down the z-axis, and the shape of the distribution can be approximated by a Gaussian function for both the z and transverse directions. The shape will be assumed to remain constant during the relatively short time it is inside the DRC and acting as a current source. Before an expression for the current density can be written down, the following parameters are defined: ne number of electrons per pulse qe electron charge Vg gun voltage me electron mass D p pulse duration parameter 3.31 wp pulse width parameter Q p pulse charge ne qe S p pulse velocity 2Vg qe me (non-relativistic) Lp pulse length parameter S p D p . In addition, a pulse normalization constant is defined as: Np Qp S p z2 2 2 exp 2 2 d dz wp Lp 0 104 . 3.32 Any waveform of constant shape traveling in the positive z-direction will be a function of ( z vt ) , where v is the velocity of the waveform. Thus, using the above defined parameters, a Gaussian shaped current density pulse traveling down the positive z-axis can be expressed in the following form: z S t 2 2 p JPulse (r , t ) zˆ N p exp . Lp wp 3.33 The constants in front of the exponential (specifically the normalization constant) have been chosen so that the integral of the above expression, over all space at a particular time, will be equal to Q p S p . This is somewhat arbitrary, but this quantity will remain constant if the pulse is compressed (or stretched) while the total charge remains constant. Since this is precisely what will happen to the pulse in the CRC, this choice of constants is appropriate. The Fourier transform is now evaluated to obtain a function of space and frequency: z S t 2 2 1 p J Pulse (r , ) zˆ N p exp exp it dt. 3.34 Lp wp 2 The factor of 1 2 is included to be consistent with the definition of a phasor. To compute the time integral, the expression inside the curly brackets must be modified in order to complete the square, giving 105 J Pulse ( r , ) zˆ N exp 2 w 1 2 p p 2 z 2 Lp 2 Sp 2 Lp 2 S iL 2 S z 3.35 iL 2 S z exp t dt. 2S 2S L 2 2 p p 2 p 2 2 2 p p 2 p 2 p p The integral is now a standard Gaussian integral and is equal to Lp S p . After some simplification, the current density transform can be written in the form: Dp 2 2 i z 2 1 J Pulse ( r , ) zˆ Dp N p exp 2 . 4 S wp 2 p 3.36 In order to match the equations in Appendix I used for calculations and plotting, this expression will be re-written in term of the ordinary frequency f rather than angular frequency ω (all expressions for the remainder of this section will also be expressed in terms of ordinary frequency): Dp 2 (2 f )2 i 2 fz 2 1 J Pulse ( r , f ) zˆ Dp N p exp 2 . 4 S wp 2 p 3.37 At this point, Equation 3.30, which is the general expression for the magnetic field in a RCCC stimulated by a current source, can be developed specifically to represent the TM010 mode field in the vacuum-filled DRC stimulated by a Gaussian current density: H ( , z ) df 010 i 2 f 010 H 010 ( ) TM TM TM N 2 f 2 2 2 4 ( f ) f i Q 0 TM 010 TM 010 106 J Pulse ( , z, f ) E ( ) TM 010 * N dV , 3.38 where the explicit spatial coordinate dependences of the various functions (for this particular mode) have been included (there is no dependence). In the above equation the subscript “N” again refers to the normalized eigenmode field patterns defined earlier. Their expressions are repeated here for convenience along with the expression for the eigenfrequency: H TM 010 N ˆi 0 TM 2.405 E010 J1 0 0 R E TM zˆ E010 J 0 010 0 N 2.405 TM f 010 , 2 0 R TM 2.405 R 3.39 where R is the radius of the cavity. The normalization phasor on the right side of the first two equations is derived from the normalization Conditions 2.67. Solving for this phasor for the TM010 mode gives: E TM 010 0 d R 0 2 J 0 2 0 0 1 2.405 R d dz , 3.40 where again R is the cavity radius and d is the cavity length. Equation 3.38 allows calculation of the TM010 mode magnetic field everywhere inside the cavity volume, when the cavity is stimulated by the electron pulse passing through it. Of particular interest is the magnetic field inside the cavity at the location of the loop antenna, which will allow calculation of the induced EMF in the loop that can then be directly measured by an oscilloscope. The loop is considered to be small enough, and the field constant enough in the loop region, that the loop location can be treated as a single point 107 on the outer edge of the cavity. The relevant coordinates for the loop, according to the previously defined cavity coordinate system, are then R and z d 2 . These coordinates can be inserted into Equation 3.38 to give a value for TM H 010 ( R, d 2) , which can then be used to calculate the magnetic flux through the loop. With the assumption that the field is constant over the small area of the loop, the flux is: TM TM 010 (t ) 0 Aloop H010 ( R, d 2, t ). 3.41 Although there are a small range of frequencies around the resonant frequency that contribute to the magnetic field, to simplify the analysis the field will be assumed to oscillate at the resonant frequency only. According to the previously defined complex notation, the above expression is then equal to (both vectors point in the ̂ direction): TM TM TM (t ) 0 Aloop 2 Re H 010 ( R, d 2, t ) exp i(2 f 010 t 2) . 3.42 010 Faraday’s Law then gives the induced EMF from the TM010 mode: TM EMF010 (t ) d TM 010 (t ) dt TM TM 2 f 010TM 0 Aloop 2 Re H 010 ( R, d 2) exp i 2 f 010 t 3.43 TM TM 2 f 010TM 0 Aloop 2 H 010 ( R, d 2) cos(2 f 010 t ). Finally, the amplitude of the EMF is the quantity that is of experimental interest: 108 TM TM TM EMF010 (t ) 2 f 010 0 Aloop 2 H 010 ( R, d 2) . 3.44 At this point, it should be discussed exactly how the above quantity relates to the actual time-domain signal that will be observed during experiment. By including only the resonant frequency in Equation 3.42, the DRC response is represented by a pure sinusoid, which of course it will not be. When a single electron pulse passes through the cavity it acts like an impulse source on a resonant system, and the response signal includes a short period of amplitude build up while the pulse is inside the cavity, followed by oscillation at the resonant frequency that gradually decreases in amplitude at a rate inversely proportional to the total quality factor of the system. The part of the response that is of interest is the amplitude of the oscillation at the resonant frequency that occurs just after the pulse leaves the cavity. Because of the large Q0 of the system, this amplitude (which corresponds to the EMF quantity in Equation 3.44) will remain relatively constant for many cycles and the signal can be approximated as a sinusoid during this time. This EMF corresponds to the stimulated magnetic field inside the DRC that is represented by the magnetic field (phasor) in Equation 3.38. This equation was derived from an analysis that assumed time-dependent current sources were pure sinusoids, and therefore produced a pure sinusoidal response. That original analysis has basically been modified to include non-sinusoidal sources, but then “pick out” the sinusoidal part of the response only. The EMF quantity in Equation 3.44 ends up being dependent on the time duration of the electron pulse passing through the DRC, which is why it is the quantity of interest. This fact allows the DRC to “detect” the pulse duration in situ without disturbing the pulse in any significant way (there will of course be a tiny reaction force on the pulse from the field it generates inside the DRC, but this effect is vanishingly small). For this reason, it is a useful tool for experiments involving pulse compression. While the DRC can theoretically detect any change 109 in pulse duration, there are two main conditions that must be met for practical use. First, the EMF voltage signal, or DRC signal, must be large enough to be detectable. Second, this signal must vary rapidly with pulse duration so the effect is noticeable above any random variation. To understand how these conditions can be met, the equations of this section must be examined in greater detail. When designing the DRC, the first choice that needs to be made is the mode of operation which will create changing flux through the loop antenna, and thus a large, detectable EMF. Due to the location and orientation of the loop antenna in the above analysis, Equation 3.41 requires the magnetic field for the chosen mode to have a significant relative amplitude at the cavity coordinates R , z d 2 and be pointing in the ̂ -direction. A quick look at the TM010 mode pattern shows it is a suitable choice. As Equation 3.39 shows, the magnetic field for the TM010 mode points only in the ̂ -direction and has no z-dependence (so there is no zero in the field to worry about at z d 2 as with many other modes). A plot of the -dependence of the field at z d 2 is shown in Figure 3-19. Here, it is important to remember that the normalization conditions are those given in Equation 2.67, which give the normalized field its strange looking units. 110 Figure 3-19 Normalized TM010 H-Field Radial dependence of the normalized TM010 H-field in a cylindrical cavity with radius 20.61 mm. With the TM010 mode chosen as a suitable mode for operation, the relationship between the loop EMF and the pulse duration can be further understood by examining the connection between the magnetic field (which directly generates the EMF) and the pulse duration. This relation is given by Equation 3.38, which can be separated into several “components” to be analyzed individually. Specifically for the TM010 mode, the main components (with units) are defined as follows: Mode Projection J Pulse ( , z , f ) E ( ) dV TM 010 Transfer Function * N i 2 f 010 TM f2 2 2 2 4 ( f ) f i Q 0 N m s TM 010 TM 010 Normalized H -Field H ( ) TM 010 N Equation 3.38 can now be rewritten as: 111 A . 3 N m 3.45 H ( , z ) [Normalized H -Field] df [Transfer Function] [Mode Projection]. TM 010 3.46 The mode projection can be represented as the volume integral of two additional sub-components; Mode Projection [Pulse Transform] [Normalized E -Field] *dVcavity , 3.47 where the sub-components are defined as Pulse Transform J Pulse ( , z, f ) TM Normalized E -Field E010 ( ) N A m2 s 1 A s N . m 3.48 Here again, the normalization conditions are responsible for the units of the normalized E-field. The mode projection can be thought of as the “overlap” of the pulse transform (at a particular frequency) and the electric field of the operating mode (in this case the TM010 mode), integrated over the volume of the cavity. According to Equation 3.46, the mode projection is then multiplied at each frequency according to the transfer function, and the integral over all frequencies gives a total amplitude which then multiplies the normalized H-field pattern for the operating mode. This finally gives the value for the magnetic field everywhere in the cavity (in standard SI units), stimulated by the electron pulse passing through the cavity. As stated previously, in order to be of practical use, the DRC must produce a signal large enough to be detected, and this signal must vary rapidly with pulse duration. The pulse transform is the only component of Equations 3.46 and 3.47 112 which affects the variation with pulse duration. All other components affect the strength of the signal. Therefore, to understand the variation with pulse duration, the pulse transform (Equation 3.36) will need to be examined in detail. In Figure 3-20, the pulse transform amplitudes for various pulse durations are plotted as a function of frequency. A single pulse is assumed to have 175,360 electrons (matching one of the values that will be used in experiments), and a velocity of 0.313c (corresponding to a gun voltage of 25 kV). Figure 3-20 Pulse Transform Variation Variation in a single pulse transform for different pulse durations. A single pulse contains 175,360 electrons and has a velocity of 0.313c. The transform functions were evaluated at cavity coordinates ρ=0 and z=0. The entire operating principle of the DRC can be derived from this one plot, along with its limitations for operation. As will be shown, because of the transfer function (related to the resonating nature of the cavity), the DRC only “sees” signals that are near its resonant frequency. For example, the DRC used in the experiments described in this document has a resonant frequency of 5.54 GHz. From the plot it is easily seen that at this frequency, the amplitude of a 20 picosecond pulse transform will be approximately seven times greater than the amplitude for a 160 picosecond pulse transform. This ratio will (almost) directly translate to a change in the detected EMF voltage signal. So, if pulses passing through the DRC are originally 160 picoseconds and are then compressed by the 113 CRC to 20 picoseconds, the magnitude of the DRC signal seen on the oscilloscope will increase by (approximately) a factor of seven, a ratio that is easily detectable. Figure 3-20 also shows, however, that at 2 GHz the same signal ratio would only be 1.2, which may not be detectable above the noise involved in the experiment. A 2 GHz DRC would be more useful for working with longer pulse durations. It can also be seen from the plot that at 25 GHz, a DRC would be most effective at detecting changes in pulses in the 10-40 ps range, but would be useless for larger pulses. So for a DRC at any fixed resonant frequency, there will be an optimal range of pulse durations to work with, and at higher resonant frequencies this range will shift to shorter pulses. Before continuing, it is necessary to investigate whether Parseval’s theorem holds for the pulse current density functions and their transforms as they’ve been defined. The theorem is as follows: 2 f (t ) dt 2 F ( f ) df where F( f ) 3.49 f (t )exp[i2 ft ]dt. Parseval’s theorem roughly states that the square of a function of time, integrated over time, must equal the square of its frequency transform integrated over frequency. The pulse current density function, as it’s defined, is a function of space and time, and the pulse transform is a function of space and frequency. Thus, to check Parseval’s theorem, the current density function (for any pulse duration) and its transform must be evaluated at the same point in space and then 114 the integrations over time and frequency carried out. Parseval’s theorem must then hold at all points in space. The pulse transform functions in Figure 3-20 were evaluated at coordinates 0 and z 0 . The corresponding pulse charge density functions, evaluated at the same point, are shown for comparison in Figure 3-21. In Appendix I, the integrations for Parseval’s theorem are carried out at several random points, for various pulse durations, and the theorem is shown to hold. This is strong evidence that the form of the pulse transform is valid. Figure 3-21 Single Pulse (Time-Domain) Time-domain functions for the pulse durations corresponding to the transforms from the previous figure. It has just been shown that the pulse transform contains within it all the information about the pulse duration. To be able to detect this information, it needs to be translated to a signal significantly large enough to be seen, for example, on an oscilloscope. Maximizing the mode projection created by the transform must therefore be the first step in creating this signal. Along with the pulse transform, the normalized TM010 E-field constitutes the other component of the mode projection according to Equation 3.47. This equation shows that the dot product of the transform and E-field vectors is integrated over the volume of the cavity to form the mode projection. Both of the vectors have only z115 components, as can be seen on their respective definitions, so the dot product is simply the product of the (complex) amplitudes of the vectors. The transform has a real and imaginary part, while the normalized E-field, as it’s been defined, is purely real. So the mode projection depends on the overlap of the z-component amplitudes of the two vectors integrated over the volume of the cavity. This integral can be separated into two parts, the integration over the radial direction ( ) and the integral over the axial direction (z). First, integration over the radial direction will be analyzed. Figure 3-22 shows the radial dependence of the two vector components that will be multiplied together. Figure 3-22 Radial Dependence of Vectors in the Mode Projection Integral Radial dependence and overlap of the pulse transform function and the z-component of the Efield. The z-coordinate of the plots was fixed at z=16 mm to show non-zero real and imaginary parts of the transform. The transform is of a 40 ps pulse. A single pulse contains 175,360 electrons and has a velocity of 0.313c. For these plots, the z-coordinate was fixed at a specific value ( z 16mm ) to show the real and imaginary parts of the transform. The gun voltage was set to 25 kV. As can be easily seen, the transform is only non-zero near the cavity axis, so for overlap to occur the E-field must also be non-zero near the axis. The TM010 mode has its maximum value for the E-field on the axis, so good overlap is ensured. This is another reason the TM010 mode is a proper choice for DRC operation. 116 Figure 3-23 Axial Dependence of Vectors in the Mode Projection Integral Axial dependence and overlap of the pulse transform function and the z-component of the Efield. The ρ-coordinate of the plots was fixed at ρ=0 where the amplitudes are maximum. The transform is of a 40 ps pulse. A single pulse contains 175,360 electrons and has a velocity of 0.313c. Next, integration over the axial direction will be analyzed. Figure 3-23 shows the axial dependence of the two vector components that will be multiplied together. For these plots the -coordinate was fixed at 0 , where the vector amplitudes are at a maximum. Since the value of Ez remains constant while the transform goes from positive to negative values sinusoidally, it is possible that the integral over this direction could have maxima and zeroes depending on the chosen cavity length, given a particular pulse velocity. This is in fact the case, as can be seen in Figure 3-24. Here, the absolute value for the entire volume integral is shown as a function of total cavity length. It is important to note that pulse transform will vary with the velocity of the pulse, which is determined by the gun voltage value, and for the above plots the gun voltage was fixed at 25 kV. At fixed velocity, various cavity lengths will correspond to various amounts of time the pulse spends inside the cavity, and this time will correspond to a certain number of periods of the oscillating field. In Figure 3-24, the orange dashed lines correspond to cavity lengths where, for a 25 kV pulse and a 5.54 GHz field, the pulse is inside the cavity for an even number of field periods. The gray dashed lines correspond to odd half-periods. 117 Figure 3-24 Mode Projection and Cavity Length Variation in the mode projection amplitude with cavity length, for a fixed pulse velocity of 0.313c. Each pulse contains 175,360 electrons and has a 40 ps duration. It is clear that maxima in the mode projection closely correspond to even periods, while zeroes in the mode projection correspond to odd half-periods. This suggests that for a fixed cavity length, variations in the gun voltage will also affect the mode projection, since a change in pulse velocity will change the transit time through the cavity. This turns out to be the case, as shown in Figure 3-25. In this plot, the cavity length was fixed at 24 mm, and the orange lines correspond again to even periods while the gray lines correspond to odd half-periods. The same relationship is seen between the mode projection and the number of RF oscillation periods the pulse spends inside the cavity. The above analysis is important since it identifies two design parameters to maximize the mode projection, namely the cavity length and the acceleration voltage (i.e., pulse velocity). 118 Figure 3-25 Mode Projection Integral and Gun Voltage Variation in the mode projection amplitude with gun voltage (pulse velocity), for a fixed cavity length of 24 mm. Each pulse contains 175,360 electrons and has a 40 ps duration. Next, it will be shown how the transfer function “picks out” a narrow range of the mode projection integral. Figure 3-26 shows the how the mode projection and transfer function vary with frequency. Here, we have shown both the real and imaginary parts of both functions, since they will be multiplied together before integrating over frequency and cross-terms will appear. The situation is simplified by the fact that over the narrow frequency range where the transfer function is non-zero (about the width of the dashed line in the mode projection plot), the values for the real and imaginary parts of the mode projection are effectively constant. 119 Figure 3-26 Mode Projection and Transfer Function Variation of the mode projection (real and imaginary parts) and the transfer function with frequency. Each pulse contains 175,360 electrons, has a velocity of 0.313c, and a duration of 40 ps. The cavity length was fixed at 24 mm. An unloaded quality factor of 9824 was assumed. The result of the multiplication gives a function (the integrand in Equation 3.46) that resembles a scaled transfer function, still narrowly centered around the resonant frequency as shown in Figure 3-27. This function is then integrated over frequency and multiplied by the normalized H-field pattern for the TM010 mode, in accordance with Equation 3.46, to give the expression for the magnetic field everywhere in the cavity. 120 Figure 3-27 Mode Projection Multiplied by the Transfer Function Mode projection and transfer function multiplied together with all parameters retaining the same values. The EMF in the loop, which will be directly measured by an oscilloscope (minus losses) can then be calculated as explained earlier. The theoretical variation of this EMF with pulse duration is shown in Figure 3-28. This largest slope in this plot is in the 40-120 ps pulse range. This means, for a 5.54 GHz cavity, it is easiest to experimentally detect changes to the duration of pulses within this range. The same conclusion was reached by analyzing the pulse transform plot (Figure 3-20) at 5.54 GHz, showing the variation in the transform is the main operating principle of the DRC. 121 Figure 3-28 Pulse Duration Dependence of EMF Variation in coupling loop EMF with pulse duration. Each pulse contains 175,360 electrons and has a velocity of 0.313c, and the cavity length is 24 mm. 3.5.2 Double Pulse Operation and DRC Calibration It has been shown that at 5.54 GHz, a DRC is best suited for detecting duration change in pulses in the 40-120 ps range. Unfortunately, the laser system used in the UEM lab produces electron pulses that are approximately 4 ps (or shorter) in duration, too short to test the DRC. In an attempt to simulate longer pulses, a Michelson interferometer set-up was introduced into the harmonic generation scheme employed to produce the UV laser pulses incident on the photocathode. An optical delay stage in one arm of the interferometer allows for the controlled generation of two time separated pulses. In this way, a 40 ps pulse can be simulated by two 4 ps pulses, their centers 40 ps apart. This “double electron pulse” can be sent through the CRC, where electromagnetic forces will change the distance between the pulse centers, either “compressing” or “expanding” the double pulse system. Figure 3-29 shows several double pulse systems used for simulating longer pulses. To determine if this type of simulation will actually work for testing the DRC, the Fourier transform for the double pulse system will need to be examined. It was established in the last section that the 122 behavior of the transform is the main operating principle of the DRC, but the transform for the double pulse system will be significantly different than that for a single pulse. Consequently, it is not at all obvious that this simulation should work. Figure 3-29 Double Pulse Systems (Time-Domain) Time-domain functions for various double pulse durations. A double pulse contains 175,360 electrons total, and has a velocity of 0.313c. Each individual pulse has a 4 ps duration. A lengthy mathematical calculation was performed in the previous section to derive the transform for a single traveling pulse. Luckily, this analysis will not need to be repeated, and a few simple properties of Fourier transforms can be utilized to achieve the desired result. The current density function for a single traveling pulse has been denoted by J (r , t ) , and its transform, calculated in the previous section, has been denoted by Ff [ J (r , t )] J (r , f ) . The current density function for an identical double pulse system can be represented by the following expression: J (r , t T 2) J (r , t T 2) , where T is the separation time between 123 the centers of the pulses. The transform can then be readily evaluated using the shift properties theorem of Fourier transforms: Ff [ J (r , t T 2) J (r , t T 2)] Ff [ J (r , t T 2)] Ff [ J (r , t T 2)] exp[i 2 f T 2] Ff [ J (r , t )] exp[ i 2 f T 2] Ff [ J (r , t )] 3.50 Ff [ J (r , t )] exp[i 2 f T 2] exp[i 2 f T 2] J (r , f ) 2cos(2 f T 2) . The end result is the original single pulse transform (for a short 4 ps pulse), modulated by a cosine function that depends on the double pulse separation time T. It is this modulation in the transform that allows the double pulse simulation to work; as the parameter T increases and decreases, simulating pulse expansion and compression, the cosine function will vary quickly enough to allow detection by the DRC. Figure 3-30 shows the transforms for several pulse separation times. It is evident that at 5.54 GHz, there is an easily detectable difference in the transforms for separation times in the 20 ps to 100 ps range. 124 Figure 3-30 Double Pulse Transforms Variation in a double pulse transform for different pulse durations. A double pulse contains 175,360 electrons total and has a velocity of 0.313c. The transform functions were evaluated at cavity coordinates ρ=0 and z=0. The rest of the analysis from the previous section still holds for the double pulses. For example, in Figure 3-31 the axial dependence of a 10 ps separation double pulse is shown. The double pulse transform has the same axial dependence as before. This means that the optimal cavity length and gun voltage values will remain the same as for the single pulse case. Figure 3-31 Axial Dependence of DP Transform Axial dependence of the double pulse transform function. The ρ-coordinate was fixed at ρ=0 where the amplitude is a maximum. The transform is of a 40 ps double pulse. A double pulse contains 175,360 electrons total and has a velocity of 0.313c. 125 In Figure 3-32, the theoretical variation of the EMF with separation distance is shown at 5.54 GHz, and it is evident that this variation closely follows the transform variation. Again, the transform is the main operating principle for the DRC. Due to the cosine function modulation in the DP transform, the EMF response of the cavity will now have a zero near a separation distance of 90 ps. This characteristic will actually make it much easier to test the DRC than is the case for a single pulse. Figure 3-32 DP Separation Dependence of EMF Variation in coupling loop EMF with double pulse duration. Each double pulse contains 175,360 electrons total and has a velocity of 0.313c, and the cavity length is 24 mm. Additionally, due to the fact that the interferometer can set the pulse separation distance with a high level of accuracy, an experiment that records the EMF response at various known separation distances can be used to calibrate the DRC for double pulse and even single pulse compression experiments. A calibration and a compression experiment for the double pulse set-up were completed and the results discussed in the following chapter. 126 3.6 Synchronization and Power When an electron pulse enters the CRC, it is critical that the EM field inside the CRC is at the proper phase for compression to occur. While the system is running, an optical delay stage is used to adjust the pulse arrival time relative to the EM field phase, and the DRC can then theoretically detect when maximum compression is being achieved. In practice, it is easier for the delay stage to be adjusted until the DRC is showing maximum expansion of the pulse (since the DRC is more sensitive to changes in larger pulses). Once the position for maximum expansion has been found, the delay stage can be moved a distance equivalent to a 180 degree shift in the phase relationship between pulse arrival and EM field. Once the optical delay stage is in the proper position for maximum compression of the pulse, the concern becomes the stability of the relationship between pulse arrival and CRC EM field phase. In some similar systems using a laser-driven electron gun and an RF compression cavity, a separate oscillator is used to drive the RF cavity. In these types of set-ups, to avoid relative drift between the oscillator driving the RF cavity and the laser oscillator, a phase locked loop (PLL) system is used to continually adjust the frequency of one of the oscillators to match the phase of the other [42]. A main advantage of the UIC UEM system is the fact that the laser oscillator not only drives the electron gun, but it also drives the RF compression cavity (CRC). Therefore, there is no concern for long-term drift between the separate oscillators and no need for a PLL to control relative drift. There remains an issue of overall drift of the laser operating frequency due to heating of the laser cavity. While this was not a problem for successful completion of the experiments discussed in this document, in the future a feedback system could be employed to continuously adjust the laser cavity length (and therefore the operating frequency) to account for the heating effects. 127 Even though long-term relative drift between the EM field phase and the pulse arrival at the CRC is not an issue, certain factors remain which cause instability to be present in the system, creating variations in the EM phase relationship from one pulse to the next. This variation causes several problems. First, some pulses may be compressed more than others, creating uncertainty in the pulse duration when it arrives at the specimen. Also, a net acceleration or deceleration can be applied to a pulse by the CRC when the phase is not perfectly matched. This will cause a variation in the location of the temporal focal point from pulse to pulse, along with a variation in arrival time of the pulse at the specimen. These types of variation mentioned are clearly detrimental to the imaging synchronization capability of the system. The main sources of instability in the UIC UEM system will now be described. The first source is the phase noise inherent in the laser oscillator. All oscillators exhibit some level of phase noise, which is closely related to the variation in an oscillator’s frequency around its nominal value. A variation in an oscillator’s frequency, of course, can be related to a variation in the time period of the oscillator. This is known as time jitter, which can be calculated from the phase noise of the oscillator and is defined as the standard deviation of the oscillator’s period. Since we are interested in the timing mismatch between the CRC EM field phase and the arrival of a pulse at the CRC, time jitter is the quantity that is of interest. A complete time jitter analysis has not been performed on the UIC UEM laser, but similar lasers have shown a level of time jitter less than 0.5 ps [43]. The second source of instability in the system, which leads to additional time jitter, is the slight fluctuation over time of the acceleration voltage in the photoelectron gun. The UIC system uses a Spellman High Voltage SL series power supply which has a voltage stability rating of 100 ppm/hour. Since the velocity of the pulse is related to the square root of the voltage and the variation 128 can be considered small, the variation in velocity will be approximately half the variation in voltage. This equates to a range in travel time from the gun to the CRC of about 0.6 ps. The third source of time jitter is from the 30 Watt RF Amplifier. As mentioned previously, a small amount of energy from the main laser oscillator, operating at 28.5 MHz, is directed towards a fast photodiode. The photodiode signal contains all of the harmonics of the laser cavity frequency up to 10 GHz. The 107th harmonic (3.055 GHz) is selected by a narrow band-pass filter, and undergoes a pre-amplification of 20 dB in order to reach the minimum input power of the 30 Watt amplifier. The UIC system uses an Ophir RF Model 5182 amplifier. The rms time jitter rating for this amplifier is 0.5 ps at 3 GHz and 24 Watts. Adding the time jitter from the three main sources of instability in the system gives a total timing mismatch of approximately 1.6 ps. It should be pointed out that for the experiments completed and discussed in this document, this level of time jitter is insignificant due to averaging over many pulses. Minimizing the effects of time jitter is more critical for imaging purposes, when a single pulse is taking an image and there is no averaging process. A detailed analysis of the effects of time jitter on imaging are beyond the scope of this document. 129 4 Experimental Results The results of six experiments related to the theory presented in this document will now be examined. The first three experiments were conducted to investigate the effects of the axial hole and the end cap holes on the characteristics of resonant cavities. The final three experiments involve a double pulse calibration of the DRC and a pulse compression experiment simulated by a double pulse set-up. The theoretical linear relationship between electrons per pulse and DRC signal strength was also verified by experiment. 4.1 Dielectric Axial Hole Effects Throughout this document, when discussing the theory of resonant cavities filled with a dielectric material, it has been suggested that the axial hole drilled through the dielectric does not significantly change the characteristics of the cavity. For this reason, the theoretical eigenmodes for a fully filled cavity with no hole have been used in calculations concerning resonant frequencies, power loss, etc. To support this critical assumption, an experiment was completed where the axial hole radius of a dielectric-filled cavity was varied, and the resulting shift in the resonant frequency was measured with a network analyzer. The goal of the experiment was to replicate the theoretical curve shown in Figure 3-2. A successful result would be a strong indication that the theory presented in Section 3.3.1 concerning the axial hole was indeed correct, and this theory then leads to the curves of Figure 3-3, which show the cavity EM field to be mostly unaffected by the axial hole. The results of the experiment are shown in Figure 4-1. To perform this experiment, a vacuum-filled aluminum cavity with end caps (using indium wire) was constructed. The cavity was designed to have a resonant frequency near 4 GHz, which corresponds to a radius of 28.71 mm. An actual YAG dielectric insert could not be used for this experiment, due to the high cost of purchasing an extra YAG insert along with the difficulty in drilling through 130 the YAG material. Instead, a much more cost effective and machining-friendly material called Rexolite was used. By comparing the resonant frequency of a filled cavity with an unfilled cavity, the Rexolite was found to have an effective dielectric constant of 2.8 at 2.5 GHz, and the Mathematica program using this parameter generated the new theoretical curve plotted in the figure. Starting with no hole at all, various hole diameters were drilled through the Rexolite, which was then inserted into the cavity housing secured with end caps and indium wire, and finally hooked up to the network analyzer to measure the resonant frequency. Each hole size tested required new indium wire and reattachment of the end caps, which by itself will cause a slight variation in the resonant frequency. The experiment was repeated 10 times at one particular hole size to provide an estimate of the variation in resonant frequency caused by using new indium wire and reattaching the end caps. This variation was found to be approximately ± 0.1%, too small to be visible at the scale of the plot in Figure 4-1. As can be seen from the figure, the experimental data closely matched the theory and clearly supports the assumption that eigenmodes for a completely filled cavity can be used in calculations. 131 Figure 4-1 Variation of Resonant Frequency with Axial Hole Size Observed network analyzer data for resonant frequency compared to a theoretical curve calculated for the dielectric material Rexolite. The resonant frequency is normalized to that of an unfilled cavity, which is 3.99 GHz. The hole radius is normalized to the cavity radius, which is 28.71 mm. Uncertainty in the data is caused by the need for new indium wire and reattachment of the cavity end caps before each measurement, but this uncertainty is too small to be seen on this scale. 4.2 End Cap Hole Effects Another possible problem with using eigenmodes for calculations is the fact that the eigenmode boundary conditions are violated by the holes in the cavity end caps. Unlike the axial hole in the dielectric, there is no exact theoretical solution for a cavity with holes in the conducting boundary to which experimental data can be compared. The end cap holes are assumed to be a small perturbation, which affect the cavity EM field in the near vicinity of the holes but have only a small effect on the overall field pattern throughout the cavity. To test this, the (vacuum-filled) aluminum cavity from the previous experiment was again used, and various hole sizes were drilled into the end caps with the resonant frequency being determined by a network analyzer for each hole size. The results of the 132 experiment are shown in Figure 4-2. The results show that there is no significant change in resonant frequency for hole radii less than 20% of the total cavity radius (the end cap hole radius for the CRC is 15% of the total cavity radius and for the DRC it is 10%). Since the resonant frequency of the cavities are not significantly changed, this supports the assumption that the end cap holes affect the field pattern only near the holes and do not have a significant effect on the overall eigenmode field pattern. Figure 4-2 Variation of Resonant Frequency with End Cap Hole Size Variation in the cavity resonant frequency when the radius of the end cap holes are varied. The radius values are normalized to the total cavity radius, 28.71 mm. Uncertainty in the data is caused by the need for new indium wire and reattachment of the cavity end caps before each measurement. This, however, does not mean that the end cap holes do not have an effect on an electron pulse passing through the cavity. The holes distort the EM field in their near vicinity, which is the region an electron pulse will pass through. Theoretical modelling [21] indicates that the distortion results in a diverging transverse electric field near the cavity entrance and exit that will affect the divergence of the pulse in the transverse direction. This change in transverse 133 pulse size will need to be compensated for by the magnetic lenses. To test the effect of the distorted fields, and the ability of the magnetic lenses to provide the appropriate correction, an experiment was conducted and the results shown in Figure 4-3. Figure 4-3 Effects of End Cap Hole Diverging Field Variation in the required magnetic lens focusing current, when an electron pulse enters the CRC at various phases (strengths) of the end cap diverging field. Uncertainty in the data is caused by the difficulty in exactly determining when the pulsed beam is focused. This experiment was performed using the CRC alone, operating at a relatively low power level of 5 W. The methodology of the experiment is based on the fact that the diverging field near the cap holes will have a sinusoidal time dependence just like the eigenmode field. Using a delay stage to control the phase of the cavity EM field when a pulse enters, the effect of the diverging field on the pulse will vary from a minimum to a maximum and back. At any particular phase, the magnetic lenses will need a specific adjustment to their operating current in order to refocus the beam of pulses. At phases where the diverging field is strong, the magnetic lenses will require maximum additional current. The AG model 134 simulations can predict this refocusing current at each phase, and a theoretical curve based on the AG model is shown in the figure. To perform the experiment, the beam was first focused to the smallest observable spot size on a YAG scintillation screen placed at the position of the Faraday cup (Figure 3-1), at one particular delay stage position, by adjusting the current in the magnetic lenses. The delay stage was then moved a distance equivalent to a particular change in EM field phase, and the beam refocused with a new magnetic lens current level. As can be seen in the figure, the experimental results agree well with the curve predicted by the AG model. The error bars of ± 2.5% are due to the obvious uncertainty involved in determining exactly when the spot size was a minimum. The results show that the effect of the end cap holes can be predicted and controlled, and therefore they do not pose an issue for successful compression experiments. It should also be pointed out that this was the first experimental verification of the predictions of the AG model developed by Berger. 4.3 DRC Signal The following experiments were conducted to test the validity of the DRC operation theory described in Section 3.5. 4.3.1 Variation of Double Pulse Separation (Calibration) The purpose of this experiment is to test the validity of the theory presented in Section 3.5.2 regarding the double pulse operation of the DRC. Verifying this theory is important since it will also verify the theory for a single pulse, and it allows calibration of the DRC for compression. In this experiment, the time separation of the two pulses in each double pulse was varied from 10 ps to 130 ps in increments of 10 ps. A low-loss 24 inch coaxial cable connected the DRC vacuum pass-through to an Agilent model N5222A microwave amplifier. The output of the amplifier was connected via another identical 24 inch cable to a 135 Tektronix model TDS6154C digital oscilloscope, where the voltage of the DRC loop EMF signal could be observed. To run the experiment, the laser amplifier was used at a pulse repetition rate of 570 Hz. At this repetition rate, after each pulse has passed through the DRC there will be enough time for the ringing signal to die off completely before another pulse enters. Therefore, the experiment can be considered a single-shot experiment and the ringing signal should match the theory (which was modeled by a single pulse passing through the cavity). A Faraday cup was used to measure the total current for each pulse train that passed through the DRC. From this the average total number of electrons in each pulse can be calculated, which is needed as an input for the theoretical model. The gun voltage was set to 25 kV. A network analyzer was used to determine the exact amplification magnitude of the microwave amplifier at the resonant frequency of the DRC, and the result was a voltage signal increase by a factor of 49.93 (33.97 dB power increase). The oscilloscope sampled the amplified voltage signal every 12.5 ps, which corresponds to approximately 14 data points per period of oscillation for a 5.54 GHz signal and 4000 total data points for the duration of the signal sampled. Because of variation in the current being recorded by the Faraday cup, averaging was employed over 5000 sets of “single-shot” data at a time. More importantly, the scope is capable of performing a FFT on the displayed and averaged timedomain data, and this information is also displayed on the screen and sent to an output file. The data from this file was used as the “observed” amplified loop EMF at the resonant frequency of the DRC. Figure 4-4 shows the results of the experiment. In the plot, four sets of data are shown. There are two sets of experimental data and two sets of theoretical values taken from the output of the Mathematica program. The first set of experimental data, referred to as “Corrected Scope Data”, is the raw data taken from the FFT calculated by the oscilloscope, with an adjustment made to 136 account for a current fluctuation that occurred during the experiment. There was a distinct drop in the mean value of the current towards the end of the experiment, from 0.015 nA to 0.013 nA. Therefore, the voltage values for DP separations of 90 ps and greater were scaled upwards by a factor of 15/13 (the relationship between current and voltage is linear). The second set of experimental data (“Cable Loss Added”) adds the cable and coupling loss in the system measured by the network analyzer as described in Section 3.3.4. In that measurement, a single 24 inch low-loss cable was connected to the DRC for analysis. The analyzer measured the return loss after the signal traveled through the cable twice, once on the way towards the DRC and once on the way back. In the current experiment, two identical 24 inch low-loss cables were used, each traversed a single time. Therefore, the loss measured previously should be identical to the loss seen in this experiment for the cables and DRC coupling. However, due to the general difficulty in accurately determining loss in microwave systems, error bars of ± 5% are added to this data set. 137 Figure 4-4 Variation of DRC Signal with DP Separation Observed oscilloscope data compared to a range for theoretical values. Cable and coupling loss measured with a network analyzer was added to the raw scope data to represent the true amplified loop EMF signal. Uncertainty in charges per pulse, loop area, and pulse charge ratio were used to create the range of theoretical values. For the theoretical data, three main sources of uncertainty were identified. While there exists some level of uncertainty in many of the parameters used in the theoretical model, the effects are insignificant compared to the uncertainty in current, measured loop area, and the DP charge ratio. The current, as measured by the Faraday cup, showed a continuous variation of ± 0.001 nA in addition to the drop in mean value towards the end of the experiment that has already been accounted for. Measurements for the loop area were made with digital calipers, but a determination of the exact size was difficult due to the irregular shape of the loop. It should be noted that since no rapidly changing magnetic field can exist inside a conductor, the only area receiving flux is the area up to the inside surface of the wire, and this is the area that was measured. The nominal figure for the loop area is 4.75×10-6 m2, with an estimated uncertainty of ± 0.1×10-6 m2. To measure the DP charge ratio, which is the ratio of charge between the two single pulses making up each double pulse system (in experiment the pulses will not be 138 exactly equal), each arm of the Michelson interferometer was blocked separately, and the resulting current generated by the single pulse coming from each arm was measured. The determined charge ratio between the pulses was estimated to be 1.2 ± 0.1. The fact that the pulses do not have equal charge causes the theoretical EMF response to have a minimum that is non-zero. A low/high theoretical boundary was then calculated by running the Mathematica program using the low/high end of the three parameter ranges (see Appendix III). The results of the experiment show excellent agreement between theory and experiment. Of the 13 experimental data points taken, 9 fall within the theoretical range when error bars are included. The (near) zero in the experimental data occurs at approximately 90-95 ps pulse separation, in agreement with the theory. 4.3.2 Variation of Current According to the DRC operation theory, the relationship between the current passing through the cavity (or, the number of electrons per pulse) and the loop EMF signal should be linear. Testing this relationship is important not only because it helps verify the theory, but also because data in the calibration experiment needed to be scaled due to current variations, and it was scaled linearly according to the theory. A simple experiment was conducted where the number of electrons per pulse (and therefore the current) was varied using beam optics, and the resulting current passing through the DRC was monitored with the Faraday cup. The corresponding loop EMF seen on the oscilloscope, adjusted for cable loss, was then plotted for each current level. Error was again estimated at ± 5%. The results are shown in Figure 4-5. The results of the experiment show that the relationship between current and signal is indeed nearly linear, as expected from the theory. The signal amplitude is also close to the theoretical values, which further supports the validity of the theory. 139 Figure 4-5 Variation of DRC Signal with Current Variation in the amplified DRC signal when the number of electrons per pulse, or equivalently the pulsed beam current, is varied. The theoretical relationship is linear. Uncertainty in the data is again caused by uncertainty in cable losses. 4.3.3 Compression In this experiment the CRC and DRC are used in tandem. First, double electron pulses are sent through the CRC at a power level of 5 W, with an initial separation set at 60 ps, and the optical delay stage is adjusted until the DRC signal is a maximum (signaling maximum compression). This means the pulses are entering the CRC at the correct phase. The compression power levels are then varied and the resulting scope signal from the DRC is recorded. The purpose of the experiment it to simultaneously test the ability of the CRC to compress an electron pulse, and to test the previous calibration of the DRC. The experiment was conducted at two different current levels, or, equivalently, at two different values of electrons per pulse, one high and one low. The purpose of this was to test if any space-charge effects could be noticed, with the assumption that the higher charge electron pulses would repel more strongly and therefore require extra power to reach maximum compression. The low 140 current value was 0.003 ± 0.0002 nA and the high value was 0.016 ± 0.002 nA. The forward power being delivered to the CRC was slightly unstable, and varied by approximately ± 0.5 W at any given data point. Error bars were added to the data to include these variations. After the initial data was taken, it was scaled linearly to match the DRC calibration current level of 0.015 nA. The results of the experiment are shown in Figure 4-6. Figure 4-6 Electron Pulse Compression Variation in the amplified DRC signal after electron pulses have passed through the CRC at various compression power levels. The experiment was conducted at two current levels to test for space-charge effects. Uncertainty in the data is caused by uncertainty in cable losses, instability in power levels, and variation in current levels. The results show that the CRC is able to successfully compress the pulses, and the DRC signal indeed matches up well with the previous calibration. At zero power the double pulses (with an initial 60 ps separation) pass through the CRC unaffected, and therefore should match the 60 ps separation DRC signal level of approximately 0.003 V. This is indeed the case. Maximum compression should correspond to approximately 0.0055, which again is what is observed. There was no significant evidence of space-charge effects, as pulses of both high 141 and low current levels reached maximum compression around 6 W of power. As the power level was raised above 6 W, the DRC signal decreases (as it should since too much power will push the pulses past each other before reaching the DRC), but the signal varies significantly more than at low power levels. This is most likely due to the individual pulses deflecting each other off axis (a “spacecharge pulse collision”) and not passing directly through each other. 142 5 Conclusion 5.1 Summary The primary goal of this research project was to explore the applicability of RF cylindrical resonant cavities to the ongoing UEM project at UIC. The initial plan for the project involved only one cavity (the CRC), which was to be added to the existing UEM column, and used to temporally focus electron pulses that were emitted from a pulsed laser-driven photocathode, and accelerated down the column through the CRC. Along the way, I proposed that a resonant cavity (the DRC) could also be used to detect the duration of pulses passing through it, without disturbing the pulse itself; in this case, for pulses of about 20 ps duration and longer. While this duration of pulse is longer than the sub-ps resolution goal of the UIC UEM project, the DRC is still useful in determining the correct phase relationship between pulse arrival and CRC EM field, and the approximate power level at which maximum compression is achieved. The underlying theory of cylindrical resonant cavity eigenmode EM fields, along with the theory of power flow to resonant cavities and sources of loss is covered in Chapter 2. Since the CRC needs to be driven by a high power RF source, the general theory of RF transmission lines and specifically coaxial cables is also presented. The process of eigenmode stimulation in a cavity by current sources, specifically current in an inductive coupling loop, is described. The concepts (and difficulties) of impedance matching a resonant cavity to a transmission line are detailed, and a theory developed from the cavity eigenmode stimulation process is used to determine the approximate size of a coupling loop that will achieve proper matching. Chapter 3 first presents the design criteria for the CRC, driven by the spacetime resolution goals of the project, with specific parameters given by the output of a previously developed computer simulation using a theoretical AG electron 143 pulse propagation model. This model simulates the electron pulse dynamics as the pulses travel down the UEM column, including the effects of RF compression. The basic operating principle of the CRC is described, and it is shown that a lowloss dielectric insert with an axial hole can be used for size reduction without significantly affecting operation. The design of the CRC is then detailed, and the process of selecting parameter values for operating frequency, acceleration voltage, field strength, etc. in order to meet the design criteria is described. Next, the construction process for the CRC is explained, including a unique indium wire sealing technique to aid in current flow and stability, as well as an alternate copper coated dielectric cavity which decreases power loss by 20%. Results of the CRC characterization process are presented, and this process was accomplished using a network analyzer to determine the resonant frequency, coupling constant, and total quality factor of the cavity. The cavity was shown to be impedance matched, and to have a quality factor high enough to allow use with a 30 W RF source. The theory of operation for the DRC is also presented in Chapter 3. This theory is basically a modification of the well-known theory of eigenmode stimulation by current sources inside a resonant cavity that have a sinusoidal time dependence. The sinusoidal time-dependent current source (such as a 3 GHz current in a coupling loop) is replaced by the current density equivalent to a single electron pulse traveling down the axis of the cavity. The Fourier transform of this current density is calculated, and the amplitude of the transform at the resonant frequency of the DRC is found to vary with the duration of the pulse, with longer duration pulses having a smaller resonant frequency amplitude. It is postulated that the driven eigenmode field amplitude will vary with the Fourier transform amplitude (at the resonant frequency) in the same way the field amplitude varies with the amplitude of a pure sinusoidal source at the resonant frequency. The DRC field amplitude can then be sampled by using the same type of coupling loop that drives the CRC. This time, however, the loop would act as 144 a receiving antenna, picking up the magnetic flux from the field inside the DRC and sending a signal through a transmission line that can be amplified and viewed on a fast oscilloscope. This signal would be directly proportional to the DRC stimulated eigenmode field amplitude. A Mathematica program was developed, based on the new theory, which could calculate the coupling loop signal given a certain pulse duration. As the electron pulses generated by the laser-driven DC electron gun have durations that are too short to effectively test and calibrate the DRC, an original double pulse experiment was designed and the additional theory for it developed. Two individual short electron pulses with a certain separation time between them have a Fourier transform amplitude that varies with the separation time. In this way, for example, a 60 ps pulse could be modelled by two 4 ps pulses with a 60 ps separation between their centers. The 5.54 GHz DRC would then be capable of detecting this separation time. Again, another Mathematica program was written to calculate the expected DRC signal given a particular double pulse separation time. Chapter 3 ends with a discussion concerning the synchronization process between the pulsed laser driving the photocathode and the EM field phase in the CRC. The electron pulses must arrive at the CRC when the EM field is at the correct phase in order for compression to occur. This phase/pulse arrival relationship must then remain stable so each pulse is compressed in the same way. Therefore, there are two basic issues to be addressed. First, there is the possibility of long term drift between pulse arrival and EM field phase. This potential problem is solved by using the laser to drive both the photocathode and the CRC EM field. The other issue is instability in the system causing random time jitter, which needs to be minimized. It was found that the high stability of the laser itself was adequate to successfully perform the necessary experiments with an acceptable level of jitter. 145 In Chapter 4, the results of experiments performed to test the theory for the CRC and DRC are presented. It was found that assumptions made about effects of end cap holes, and the axial hole in the CRC dielectric, were correct. One of the experiments, concerning the diverging electric field caused by the end cap holes, also provided some verification of the AG model computer simulation. A calibration experiment for the DRC was successfully performed that strongly supported the double pulse theory. Using the calibration data, a compression experiment using the CRC and DRC together was performed, which showed maximum compression occurring at 6 W of RF power. The linearly scaled DRC signal data for this experiment matched the previous calibration data. 5.2 Future Prospects There are many possible applications for RF and microwave resonant cavities in the field of UEM which warrant further study. The addition of dielectric material not only allows for size reduction as shown in this thesis, but also creates the possibility to operate cavities in modes which have a z-dependent field, such as the TM011 mode. The typical problem that occurs when using these types of modes is that the pulse must travel through the cavity at the phase velocity of the z-dependent EM field in to achieve proper compression. In a vacuum-filled cavity this phase velocity is the velocity of light, but in a dielectricfilled cavity the phase velocity is lowered by a factor equal to the square root of the dielectric constant. A YAG-filled cavity, for example, would have a phase velocity of roughly c 3 , or about the velocity the pulses are accelerated to with the UIC gun at 25 kV. Also, by using the TM011 mode there would be a reduction in operating power, which reduces any heating related effects. In addition, the issue of the diverging electric fields at the end cap holes would be greatly reduced, since the TM011 eigenmode field is zero at the cavity ends. Another area for possible research involves the copper coating process and machinable dielectrics such as Rexolite. These dielectrics could be formed into 146 shapes whose eigenmodes can’t be directly solved for, but could be approximated (and designed) with computer software. The possibilities for creating advantageous field patterns to reduce power while increasing compression (or acceleration) would be endless. 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IEEE Journal of Selected Topics in Quantum Electronics, 1998. 4(2): p. 179-184. 150 Appendix I Mathematica Code for DRC Single Pulse Analysis Presented here is the Mathematica code that generated the plots used in Section 3.5.1, showing the theoretical analysis for the EMF response of a single pulse traveling through the DRC. 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 Appendix II Mathematica Code for DRC Double Pulse Analysis Presented here is the Mathematica code that generated the plots used in Section 3.5.2, showing the theoretical analysis for the EMF response of a double pulse system (individual pulses identical) traveling through the DRC. 168 169 170 171 172 173 174 Appendix III Mathematica Code for DRC Unequal DP Analysis Presented here is the Mathematica code that generated the plots used in Section 4.3.1, showing the theoretical range of values for the double pulse (individual pulses not identical) calibration experiment for the DRC. 175 176 177 178 179
