THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN QUARTERLY PROGRESS REPORT NO. 3 FOR BASIC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONICS This report covers the period November 1, 1963 to February 1, 1964 Electron Physics Laboratory Department of Electrical Engineering By: H. K. Detweiler?A'pprVed' by: 6. M. E. El-Shandwily; Yeh C. Yeh,' B. Ho E. Rw,,:: Project Engineer J. E. Rowe.-,-., ~ C. Yeh ApproVed by:. ~. 0 J E Rowe, Director'Electron Physics Laboratory Project 05772 DEPARTMENT OF THE NAVY BUREAU OF SHIPS WASHINGTON 25, D. C. PROJECT SERIAL NO. SR0080301, TASK 9391 CONTRACT NO. NObsr-89274 March, 1964

-42 ( -0-11^'( -1, k- atf e - 6 ~v, c~v~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

ABSTRACT Generation and Amplification of Coherent Electromagnetic Energy in the Millimeter and Submillimeter Wavelength Region An experimental low-frequency model of the frequency multiplier has been designed. A scheme of feedback to enhance the transfer efficiency is included. The tube is in the final stage of assembly and alignment. Difficulties in loading a helix designed to operate at 30 Gc into a BeO tube have not been fully resolved. Experimental data indicates that a material having a higher resiliency than tungsten should be used for the helix. Analysis of Amplitude-and Phase-Modulated Traveling-Wave Amplifiers The analysis of the operation of the TWA with a multi-frequency input has been extended to account for the multivalued nature of the electron velocity in the beam. Boltzmann's transport equation is used to describe the kinematics of the electron beam. Study of a D-c Pumped Quadrupole Amplifier A comprehensive study of the coupling mechanism between the various modes of operation of a d-c pumped transverse wave device has been made by means of the coupled-mode theory. Equations for the gain for the various modes and different pump fields are computed and compared. - i i

TABLE OF CONTENTS Page ABSTRACT iii LIST OF ILLUSTRATIONS vi LIST OF TABLES viii PERSONNEL ix ARTICLES PUBLISHED DURING THE LAST QUARTER x 1. GENERAL INTRODUCTION 1 2. GENERATION AND AMPLIFICATION OF COHERENT ELECTROMAGNETIC ENERGY IN THE MILLIMETER AND SUBMILLIMETER WAVELENGTH REGION 2 2.1 Study of the Interaction Between an Electron Beam and a Dielectric Circuit 2 2.2 Study of Frequency Multiplication in an Angular Propagating Circuit 2 2.2.1 Construction of the Experimental Tube 2 2.2.la The Electron Gun 4 2.2.lb The Input Coupler 4 2.2.1c The Multiplier Cavity 4 2.2.1d The Feedback Coupler 5 2.2.le The Collector 5 2.2.2 Future Work 5 2.3 Investigation of High-Thermal-Conductivity Materials for Microwave Devices Above X-Band 5 2.3.1 Introduction 5 2.3.2 Experimental Effort 6 2.3.3 Future Work 7 3. ANALYSIS OF AMPLITUDE AND PHASE-MODULATED TRAVELINGWAVE AMPLIFIERS 8 3.1 Introduction 8 3.2 Method of Analysis 8 3.3 Solution of the Equations 18 3.4 Program for the Next Quarter 40 -iv -

Page 4. STUDY OF A D-c PUMPED QUADRUPOLE AMPLIFIER 40 4.1 Introduction 40 4.2 Coupled-Mode Equations of Beam Dynamics 41 4.3 Coupled-Mode Analysis of D-c Pumped Quadrupole Amplifiers 43 4.3.1 General Procedure of the Analysis 43 4.3.2 Staggered Quadrupole Amplifier 46 4.3.3 Other Pumping Fields 60o 4.3.3a Twisted Quadrupole Pump Structure 60 4.3.3b Periodic Ring Quadrupole Structure 62 4.3.3c Electrostatic Slot Pump Field Structure 64 4.3.4 Gain Computation of the D-c Pumped Amplifiers 65 4.4 Future Work 73 5. GENERAL CONCLUSIONS 73

LIST OF ILLUSTRATIONS Figure Page 2.1 Assembly Drawing of Cyclotron-Frequency Multiplier. 3 4.1 Geometrical Configurations of a Staggered Quadrupole Pump Structure. 47 4.2 Phase Constant vs. Pumping Parameter M for the Fast Cyclotron Wave in a Staggered Quadrupole Pump Structure. 0q = 2pc, Fast and Slow Cyclotron Wave Interaction. 52 4.3 Amplitude Gain vs. Pumping Parameter M for the Fast Cyclotron Wave in a Staggered Quadrupole Pump Structure. qC = 2, Fast and Slow Cyclotron Wave Interaction. 553 4.4 0)-P Plot of the Cyclotron-to-Synchronous Wave Interaction in a Staggered Quadrupole Pump Structure, Passive Coupling. 57 4.5 w-P Plot of the Cyclotron-to-Synchronous Wave Interaction in a Staggered Quadrupole Pump Structure, Active Coupling. 58 4.6 Phase Constant vs. Pumping Parameter M for the Fast Cyclotron Wave in a Staggered Quadrupole Pump Structure. q = pc Fast Cyclotron-to-Synchronous Wave Interaction. 59 4.7 Geometrical Configurations of (a) Twisted Quadrupole-Type, (b) Periodic Ring-Type and (c) Periodic Slot-Type of Quadrupole Pump Structures. 61 4.8 Gain vs. Pump Voltage for Cyclotron-toCyclotron Type of Interaction in Different Pump Fields for V = 100 Volts, 5 a = 1, and n = 4. 67 4.9 Gain vs. Pump Voltage for Synchronous-toSynchronous Type of Interaction in Different Pump Fields for V = 100 Volts, D a = 1, and n = 4. c 68 4.10 Gain vs. Pump Voltage for Cyclotron-toSynchronous Type of Interaction in Different Pump Fields for V = 100 Volts, B a = 1, and n = 4. c 69 -vi

Figure Page 4.11 Gain vs.: a for Synchronous-to-Synchronous c Type of Interaction in Different Pump Fields for V /VO = 4, L = 40 cm and U = 4.2 x 106 m/se,c. 72 -vii

LIST OF TABLES Table Page 4.1 Gain Equations for Various Types of' Beam Interaction and Different Types of Pump Fields 70 -Vill -

PERSONNEL Time Worked in Scientific and Engineering Personnel Man Months* G. Hok Professor of Electrical Engineering.30 C. Yeh Associate Professor of Electrical Engineering 1.16 M. El-Shandwily Research Associate 1.36 H. Detweiler Research Assistants 1.36 B. Ho 1.56 Service Personnel 5.27 * Time Worked is based on 172 hours per month. -ix

ARTICLES PUBLISHED DURING THE LAST QUARTER C. Yeh, J. C. Lee, "Feedback in Cyclotron-Wave Frequency Multiplier", Proc. IEEE, vol. 52, No. 3, p. 314; March, 1964. -x

INTERIM SCIENTIFIC REPORT NO. 5 FOR BASIC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONICS 1. General Introduction (C. Yeh) The purpose of this project is to investigate new ideas in the area of microwave devices and quantum electronics. The program is envisioned as a general and flexible one under which a wide variety of topics may be studied. At present, the following areas of investigation are in progress: A. Study of frequency multiplication in an angular propagating circuit. A tube based upon the analysis described in the preceding reports, No. 1 and No. 2, is designed. The construction of this tube is nearly completed. B. Investigation of high-thermal-conduct^ivity materials for microwave devices above X-band. The work on developing a technique to braze a 30 Gc helix to a BeO tube is continuing. C. Analysis of amplitude- and phase-modulated traveling-wave amplifier. The theory has been extended to cover the possibility of having multivalued functions of electron velocity in the beam. D. Study of a d-c pumped quadrupole amplifier. A comprehensive study of the d-c pumped transverse wave device has been made by means of the coupled-mode theory. Some interesting aspects of the coupling mode are discussed. The work on the study of interaction between an electron beam and a dielectric circuit has been temporarily suspended.

-22. Generation and Amplification of Coherent Electromagnetic Energy in the Millimeter and Submillimeter Wavelength Region 2.1 Study of the Interaction Between an Electron Beam and a Dielectric Circuit (G. Hok) The work on this task has been temporarily suspended because of shortage of available manhours. 2.2 Study of Frequency Multiplication in an Angular Propagating Circuit (C. Yeh, B. Ho) 2.2.1 Construction of the Experimental Tube. The design of the experimental frequency multiplier tube using a multipole cavity as the multiplier element and a feedback scheme to enhance the conversion efficiency was presented in Quarterly Progress Report No. 2. During the present period, progreas has been made in the actual construction of such a tube. The design data are as follows: Input frequency 696.3 mc/s Output frequency 2785.0 mc/s Magnetic flux density 250 gauss Beam voltage 20 volts Beam current 250 microamperes Maximum r-f beam power 770 mw Maximum -beam cyclotron radius 17/32 inch Input coupler plates 2-1/4" x 9/16" Input coupler separation 9/16" Feedback coupler plates 1-15/16" x 31/64" Feedback coupler separation 31 64" The detailed diagram of the assembly of the cyclotron frequency multiplier is shown in Fig. 2.1.

-50 0 ~~~~~~~ ~~~o L)~~~~~~~~~~t o~0 15 0 z - o D. w 0 CL z~~~~~~ F~~~~~~~~-b z~~~~~~ r3 o w O~~~~~~~~~~~~~~~~~~~C H ~ 01 o~ff I z~~~~~~~~~~~~~~~~~I -l~~ ~ -- NW ~ \N~-~C 0 11~~~~~~~~ cr~~~~~~~~r -J: 0~~~~~~~~~a D~~~~ Hg::~~~~~a ~'~~~~~L ~i'~~~~~~~~~~~~~0-

-42.2.la The Electron Gun. The electron gun used in this design is a conventional cathode ray gun with the second anode section removed. Since it is to be used as a low voltage gun, it may be necessary to apply a slightly positive potential to the control grid in order to get sufficient emission from the gun. The remaining parts of the tube, i.e., the coupler plates, the cavity, the feedback plates and the collector, will all be connected to the same d-c potential as the first anode. The focusing of the beam is helped by the magnetic field. 2.2.lb The Input Coupler. The input coupler consists of two pairs of mutually perpendicular Cuccia couplers. They are all connected to the same d-c potential as the first anode of the gun, but are excited, r-f wise, from two separate sources. This scheme can be achieved by a network of r-f chokes and d-c blocking capacitors, which is not shown in the assembly drawing. The plates are held together by ceramic insulators and fit into the glass wall by finger springs. The centers of the coupler are carefully aligned with the gun on one end and the cavity on the other end. One r-f source, the lower frequency input source, is connected to one pair of the couplers while the feedback source is connected to the perpendicular pair. The leads are brought out through the pins on the base. 2.2.1c The Multiplier Cavity. The multiplier cavity is adapted from a hole-slot-type cavity of an S-band magnetron. The cathode of the magnetron is removed to make room for the whirling cyclotron wave beam. The magnetic pole pieces are replaced by metal rings which extend on both sides to make connections for the couplers and the rest of the tube. The cavity has a resonant frequency of

2785 mc/s. It has a total of N = 8 holes. The tips of the slots are strapped so that the desired 1-mode is well separated from other modes. The output, at frequency (N/2) w is taken from the cavity by a c coupling loop located in one of the holes. The output connector is modified from the original design by bending it toward the cavity so that the entire structure may be inserted in a long magnetic coil three inches in diameter. The cavity will again be biased at the same d-c potential as the first anode of the gun. 2.2.ld The Feedback Coupler. The feedback coupler consists of a single pair of plates of similar design to that of the input coupler but modified in dimensions. The leads for these plates feed through the pins on the collector side of the tube. This facilitates making the feedback connections with the input coupler outside of the tube with any desirable amount of delay inserted. The d-c potential on the plates is again the same as that on the first anode of the electron gun. 2.2.le The Collector. The collector is simply a flat metal disc. It is maintained at the same d-c potential as the anode. Since the electron energy will be low, no special problems arise in the design of the collector. 2.2.2 Future Work. It is expected that the final assembly of the tube will be completed during the next period. Experiments with the tube can begin immediately. 2.3 Investigation of High-Thermal-Conductivity Materials for Microwave Devices Above X-Band (H. K. Detweiler) 2.3.1 Introduction. Further efforts have been made during this period to develop a satisfactory technique for brazing a tungsten

-6helix designed for operation at 30 Gc into a smooth-bore BeO tube. The present effort has been directed toward finding a means for loading the helix into the tube in such a way as to obtain uniform contact. A detailed account of the work performed is given below. 2.3.2 Experimental Effort. At the conclusion of the previous quarter a procedure for preparing the helix for brazing and a satisfactory brazing cycle had been established. However, a serious difficulty had been encountered when attempts were made to load the helix into the tube. The loading technique was to first wind the helix on a mandrel so that the helix outside diameter exceeded the inside diameter of the tube by about 2 mils. After firing the helix to set it, it was wound on a smaller size mandrel so that its O.D. was about 2 to 2-1/2 mils smaller than the tube I.D. This amount of clearance was necessary so that the titanium coating on the helix would not be rubbed off during loading. It was then inserted in the tube and released. When this was done it was found that the helix did not spring back enough to make good contact with the tube. A program was undertaken to determine the firing cycle which would give the desired amount of spring to the helix in order to ensure good contact. The first series of tests involved firing the helices in a hydrogen atmosphere at 10500C for periods of time ranging from 25 minutes to 1-1/2 hours. In each case the helices were found to have insufficient spring-back; the helices increased on the average only 1-1/2 mils in diameter when released, whereas at least 2-1/2 mils is required. In the next series of tests the helices were fired for long periods of time, i.e., 1 to 1-1/2 hours, in hydrogen at lower temperatures and allowing them to cool with the furnace. Again the helices failed to spring back a sufficient amount.

-7Next, vacuum firing of the helices was attempted. When fired at temperatures between 8000C and 10000C for 15 minutes they were brittle and broke during attempts to wind them on the smaller diameter mandrel. When fired at temperatures below 8000C the helices did not break but as before they did not spring back enough when released. The most recent test consisted of firing in hydrogen at 1150~C for 20 minutes. Of three helices fired in this manner, one broke when being wound down and the other two did not spring back enough when released. It can be seen from the outcome of the above tests that the loading difficulty has not been resolved. The results of these tests seem to indicate that, when the point is reached where the helices are no longer so brittle that they break when being wound down, they do not possess sufficient resiliency to spring back more than about 1-1/2 mils. At this time, this particular approach to the problem does not seem to offer much hope of success. 2.3.3 Future Work. Work will be continued to develop a helix loading technique. An investigation will be made into the possibility of finding another suitable helix material which has a higher resiliency than tungsten. Attempts will also be made to obtain BeO tubing with greater dimensional uniformity so that less clearance between the helix and tube is required for loading. Pending successful solution of the loading problem, d-c heat tests of the power handling capability and r-f cold tests of the electrical characteristics of the brazedhelix structure will be conducted.

-83. Analysis of Amplitude and Phase-Modulated Traveling-Wave Amplifiers (M. E. El-Shandwily, J. E. Rowe) 3.1 Introduction. In the previous quarterly progress reports, an analysis was made of the operation of the TWA with a multi-frequency input. In this report, the problem is treated by another method. The reason is that the previous analysis was based on Pierce's1 theory in which it was assumed that at any plane z = constant the electron velocity is a single-valued function. Even if the initial thermal velocity spread is neglected, the crossing of the electrons when the beam is bunched makes that assumption invalid. In order to account for electron crossovers (multivalued velocity function), the kinematics of the electron beam will be described by the Boltzmann transport equation2. This approach has been applied by Watkins and Rynn3 to study the effect of the initial velocity distribution on TWT gain, by Kiel and Parzen4 to study the nonlinear interaction in the TWT and by others to study the noise propagation on electron beams. 3.2 Method of Analysis. To reduce the calculations to a reasonable amount the following assumptions are made: 1. The gain parameter C is small compared to unity. 2. The tube is long enough so that at the output only the largest growing wave is considered. 1. Pierce, J. R., Traveling Wave Tubes, D. Van Nostrand Co., New York, N. Y.; 1950. 2. Rose, D. J. and Clark, Jr., M., Plasma and Controlled Fusion, M. I. T. and Wiley, N. Y., p. 58, 1961. 3. Watkins, D. A. and Rynn, N., "Effect of Velocity Distribution on TWT Gain", Jour. Appl. Phys., vol. 25, pp. 1375-1379; November, 1954. 4. Kiel, A. and Parzen, P., "Nonlinear Wave Propagation in TWA", IRE Trans. PGED, vol. ED-2, pp. 26-34; October, 1955.

-93. There are no space-charge forces (QC = O). 4-. There is no loss. 5. The collision between electrons and gas molecules is neglected. 6. One-dimensional analysis. The circuit model used is that of Brillouin5, and the electronic equations are replaced by the Boltzmann transport equation. Therefore the two working equations can be written as follows: The circuit equation is a2V(z,t) v2 a2V(z,t) = z v 62p(z,t) 6t2 a2 0 0 z and the Boltzmann equation is aF(z,u,t) + u aF(z,u,t) + aV(z,t) aF(z,u,t) 0 (32) at az az u where p is the beam charge density and is given by 00 p = - j F(z,u,t)du.(33) — 00 V(z,t) is the circuit voltage, v is the-phase velocity of the circuit voltage, 0 Z is the circuit impedance, F(z,u,t) is the density function which gives e times the number of electrons between z, z + dz with velocity between u, u + du per unit beam cross section, = e|/m, the magnitude of the charge-to-mass ratio of the electron. 5. Brillouin, L., "The Traveling Wave Tube", Jour. Appl. Phys., vol. 20, pp. 1196-1206; December, 1949.

-10To solve Eqs. 3.1 and 3.2, the boundary conditions must be specified. In this analysis it will be assumed that the input consists of two signals with angular frequencies, w. 1 2 V(O,t) = B cos w t + B cos o t 1 1 2 2 jw t - t w t j(3.4a) 1 2 -jc o ~ 1 2j 2 1 B e + e 2) + B (e2 + e2 ) (3.4a) 2 1 +f 2 The voltage should behave initially as -CO z - z B e 1 cos(c t - 3 z)+ B e?cos(c t - B z), 1 1 1 2 2 2 where = cu /v, = co /v. Assuming that a, K, c <<, gives 1 1 1 2 22 1 1 2 2 the following for av(z,t)/az: V (-z, t) = B B sin co t + 1 B sin co t az z=o 1 1 1 2 2 2 jC t -jc t J) t -jw t = 2B( 1 - 2 e 2 2 B jec ) 22 2 (13. 4b) At the input to the tube, the beam is assumed to be unmodulated and neglecting the thermal velocity the density function becomes F (O,u,t) = a 6(u-uo), (3.4c) where u2 = 2e/m V, V is the d-c beam voltage and 6(u) is the Dirac 0 0 0 delta function. Also, F(z,u,t) -~O as u -0O, u -,. (3.4d)

-11The problem is to solve Eqs. 3.1 and 3.2 subject to the boundary conditions (Eqs. 3.4). It is to be noted that although the circuit equation (Eq. 3.1) is linear, the Boltzmann equation is not. Therefore, the density function and the circuit voltage will contain, in addition to the input frequencies and their harmonics, all possible combinations of frequencies. The circuit voltage and the density function will be written in the following general forms: m0 s w (s,w) j(nw + mw )t V(z,t) = B B Vnm e 1 2 (3-5) s,w=o n=-s m=-w m0 s w (s,w) j(mo + mr )t F(z,u,t) = B B1 B2 Fm e 1 2 (3.6) s,w=o n=-s m=-s (S,w) (s,w) where V is a function of z and F is a function of u and z. n,m n,m The expansion in the input voltages B and B is valid provided 1 2 that these quantities are small, which is usually assumed in solving nonlinear differential equations6. The harmonic expansion in the input frequencies is stopped at Inj = s, Iml = w, since there can be no harmonics higher than the nonlinearity. Also, since the voltage should j t jL t behave initially as B e 1 + B e 2,then n + s and m + w must be 1 2 even. Equation 3.1 was derived originally for a single driving frequency. This is done by replacing the helix by an equivalent transmission line with current injected along its length. Since the equation is linear, then for every charge density component on the beam there will be 6. Minorsky, N., Non-Linear Mechanics, Edwards Brothers, Inc., Ann Arbor, Mich.; 1947.

-12a circuit voltage of the same frequency. Therefore Eq. 5.1 is valid for the general case of a multi-frequency input, where the circuit voltage has the general form of Eq. 3.5. Equation 3.2 is valid for any distribution function. Substituting Eqs. 3.5 and 3.6 into Eq. 3.1 gives: 5X s ~ w ~ ~(s,w) j(rvo +mo )t V (r T +mo )2 BS BW V 1 2 1 2 1 2 n,m s,w=o n=-s m=-w w (sw) (s,w) nnm 1 2 dz2 s,w=o n=-s m=-w oo s w 00 / \ s,w=o n=-s m=-w O j(r= + nm )t *e 1 2 (537) where v is the phase velocity at frequency 1n= + m 2| and Z is n1m 1 2 Equating powers of B Bw on both sides, and noting that the 1 2 exponentials are orthogonal functions, then one can equate the coefficient of the exponential term on each side of Eq. 3.7 to get (S,w) d2V(S,w) + )2 V 2 n,m (1121 o2) Vnm n,m dz2 0 (s,w) - -Z v (rt I + Bs)2 B F du (5.8) _nm nm 1 2 J n,m 0 Substituting Eqs. 3.5 and 3.6 into Eq. 3.2 yields

00 (s,w) j(rn + nSo )t _ X j(nwo + nm) B BW F e 1 2 1 2 az 1 2,1 2 n,m s, w= o n=-s m=-s m=-w (s,w) (s w dF ij(r n + m u )t +1 S n e s,w=o n=-s m=-w 00 s w (S,w) B( B dV. mj(n"~ + me )t2 s+ w=o n "=-s m"=-w hand side of the above equation can be written as c ~o s w n=n-+s m'+BsX X X E X B BB S,w=o sW=o =o m=m'=-s m' =-w+ Using the Cauhy produt and letting m n " + n = n the rightThe index of summation can be arranged such that the following holds: o s w +(sw) j(n=n + s- wL I L 1 2) 1 2 n 2m e 12 2 s,w=o n=-s m=-w cavo aF, j(srk + xrm )t n-n m-m n,m 1 2 (sVw) j(srz + n-o )t n,m j(r~ + n~ )t

-14Equating the coefficient of the exponential in the above equation gives (s,w) (s,w) 6F (r + ) F + n,m 1 2 n,m v v, a6F( s-T,w- ) -z ~1~ 1~ n-n,m-m' n',m' (39) Z L, az 6u r=o 5=o n'=-s m'=-w It is found that it is more convenient to write V(z), F(z,u) in terms of the slowly varying variable 0 = 5Cz. (s,w) (s,w) J unm V (z) V' ( ) e nm n,m n,m 1 nm (s,w) (s,w) -j F (z,u) F' (D,u) e (3.10) n,m n,m 1 where - C z and =w /u i e1 1 o The introduction of these variables will reduce the amount of algebra and facilitate the subsequent calculations considerably. Using Eq. 3.10, Eqs. 3.8 and 3.9 reduce to ( s,w) (s,w) (S,w) d2V' dV'n,m (rzo + m )2 V' (() + v2 B c2 m - 2 JC nm (S,W) (s,w) nm n,m nm 2 j n,m 0 and

15-m )F' ( snw) j(D + m) Fn (S,W) (u,u) + u [C nm (s,w) 2 n,m nm n, s w s w dVI dV q=o g=o n'=-s m'=-w ( s-j,w- ) nn',, lm (3.12) A n-nm-m n-n am-mm A convenient way to solve the above two equations is to use the Laplace transform7 as given in the following definition: (3.13a) LL [FA,m (',u)j ff Fs1 (O,u) e(6+pu d~du nm n,m 0 0 - F (6,p), Re (6) > a, (3.13b) n,m 2 where a, a are some constants which make the integration in Eqs. 3.13a 1 2 and 3.13b converge. Taking the Laplace transform of Eq. 3.11 and using the boundary conditions gives 7. Churchill, R.V., Operational Mathematics, Mathematics cGraw-Hill, New York; 1958.

-162(S, + )22 I()v c (( ) - 5(sl) b(w,o) ( 1- +,2)2 V( ) (s) + V2 p2C2 52 n,m ) F 2F (sl) 5(wYO) 1 2 n,m nm nm 1 1 - 2 (s,O)5(w,l) + j blo C1 (s,l) 5(w,O) 2 1,190 1 +1 5(w) b(s) v2 2 j C nm s,V () - oi oi(sn) 2 rim n,m 6(s,l) (w,O) (sO) (w,l) V2 2 (5) 2 6(wO) 2 - nm nm n, m 2(s,w) - V z (nD + )2 (s,o). (3.14) nm nm 2 nm The positive sign is for n = 1, while the negative sign is for n = - 1. Taking the double Laplace transform of Eq. 3.12 and using the boundary conditions yields (s,w) _(s,W) ~ (s,p) =( s,w) -u OP aF n m (6,p) - au e (s,0O) o(w,O) + j nm'm C+j S W S - 2- j I- I I I6,,r) (5-r) 23t j n_n ~m-m c-joo =o =o n =-s m-w 2 5(s,l) 6(w,O) PC - I 6(s,O) 5(w,l) PC - j B V~~~~

-17where - Re(6), 6(n,m) = O, n f m = 1, n =m C - Re(r), a < C < -. 1 2 In writing Eq. 3.15 the complex convolution formula has been used to write the right-hand side8. The condition on the real part of r in Eq. 3.15 defines a strip in the r plane for the integration of the right-hand side. Only the poles of F(r,p) will contribute in evaluating the integral. This is because the path of integration always lies to the right of the poles of F(r,p) and to the left of the poles of the bracketed term. Therefore, by closing the contour to the left, the value of the integral over the semi-circle will vanish if F(r,p) and the bracketed term are proper rational fractions (which will be assumed), and the integral gives X[{mmn-n m-m' n-n',m- m' 2 (s,)(w,)c 1 (s_(S-,w- ) -( S-,w- ) 2 6(w,l) Cj ~F,m, (r,p) at the poles of F t (r,p) Rearrange the terms in Eqs. 3.14 and 3.15 and rewrite in the following final form: 8. Aseltine, J. A., Transform Method in Linear System Analysis, App. A-7, McGraw-Hill, New York; 1958.

-18(S W)[ ( r)[( + mno )2 + V2 (p2 C2 62 _ 2j CnmO - 2nm) + nm V - [ jb0o C10 (s,) low,) +- 5~nm L jb c 03 Ca + 2 j PC DnmC nm nm 1 2 m and (s,w) -uoP aF (),p) j(rn + no) _(s,w) $Cau e n,m p+ ( s (6P) - 6(s,0) 5(w,0) ap - nm j -C nmC JnmC C+joo S W S W j B -g C6 2irc mL L L i nm c-joo 7r=o 5=o n'=-s m'=-w L{C(b-r) - Jn-nl,m-m' }V-n'im-m' (6-r) - - PC {(s,1) 5(w,O) + 6(s,0) n(w,l)j n',m's (r,p) dr.(317) In the above equations PC is lo0 Clo0 3.3 Solution of the Equations. The starting point in solving Eqs. _(o,o) 3.16 and 3.17 is to let s = 0, w = 0. From Eq. 3.17 one gets F (6,p). _( o,o) By knowing Foo (6,p) and putting s = 1, w = O in Eqs. 3.16 and 3.17, then 0,0

-19_(1, o) (1,o) it is possible to get F1,0 (6,p) and V 10 (5) and repeating for s = 0, w = 1 to get F O)(,p) and VO )(b), which are the first-order approximations. From the first-order solution the second-order approximation can be obtained and the process is repeated to get higherorder approximations. This process is carried out as follows: From Eq. 3.17 for s = O, w = 0 _(o, o) F (5,p) au -u p +- e O ap therefore, _(o,o) -u p (o,o) a 0 F (6,p) = - e(.18) 0,0 (3.18) First-Order Solution Let s = 1, n = + 1, w = 0 in Eq. 3.17 ( 1,0) a,po (p + p)jw p c+j 6P jp - C b'1, j -C b 2ij c- o _( (lo) solving for F l, (5,p) yields.1.o) 1 -pu 1 1,1y~~~~o o _(i,o) qa[(Cob-j) V) (s) - a [puo C- o - j + C1o6] e (Uo C o6)2 (3-19) For s = 1, w = O, n = 1, the circuit equation (Eq. 3.16) gives:

-20(1,o) v2 V o () [c2+v2 (%2C252 _ 2j p2C6 _ p2)] + 10 [(-6+jboC ) 2c2 10 21 10 2o (1,o) + 2j2C] = - Z v Fw2 (6,0). (3.20) 10 10 1 10 Let p = 0 in Eq. 3.19 and then substituting into Eq. 3.20 and arranging terms one gets: (1,o) V (6) I O raZ1o v COc p2 (-j + c 5) + C3O 522 [-5C +jb C +2j] 10 2 C2 62 (2C b -2jC 6+C2 b2 +C2 52) 2 C2 010 10[10 10 o10 10 o10 v (3.21) At this point some useful definitions will be introduced. Pierce's gain parameter C3 = ZmIo/4Vo, where Z is the circuit nm nm o o nm impedance at frequency (nro + mu ). Also, from the definition of the 1 2 density function it follows that I'= rF udu = au 9C~ ~o Introducing this definition into the definition of C gives n,m qaZ C3 = nm (3.22) nm 2u Substituting from Eq. 3.22 into Eq. 3.21 and neglecting terms of the order of C compared to 1 yields ( 1,)`2 V1 (6) = (3.23) 2j [62(-j6+blo)+1] It is seen that when the denominator is equated to zero, it gives the determinantal equation of the traveling-wave tube under theassumptions made.

-21It will be assumed that the tube is long enough such that only the wave which corresponds to the pole with the largest real positive part is present at the output. Taking the inverse Laplace transform of Eq. 3.23 and using the definition Eq. 3.10, one finds for the first-order approximation for the voltage with frequency w (1,0) 82 e 1 1 1 j(ao t - B z) V (z,t) = 1 e 1 1 (3.24) 2j(6 - ) ( -6 ) 1 2 1 3 From the original definition of the voltage it is seen that the voltage (1,0) is given by 2Re V (z,t). Substituting Eq. 3.23 into Eq. 3.19 and making the same approximations, one gets -pu (l,o) qa [puO C lo-j] e 1o0 2u2 C2 [62(_j 6 + bl) + 1] 0 10 10 To obtain the first-order solution for the signal with frequency w, the above procedure is repeated for s = O, w = 1, m = 1, and the 2 result is o2 c2 10 10 2 (0,1) 12 C2 v (6) 1 O CC,01 (.26)1 V, 1 (o) C 2 C Ca' (3.26) 010101 11010 2. 101 10 10 L 01 01 01 01 2j o 5a L c + b + 1 -pUopulolo C (o,1) e l O1l o0 1 (b,p) = 2ua0 10 10Ca 2 01~ BC C 22 C o (3.27)

-22Assuming that the circuit is nondispersive in a certain frequency band and that w and w are within this band, then b = b. Taking the 1 2 10 01 inverse Laplace transform of Eq. 3.26 one obtains (0,1) 62 e (01 01 1 t-B z) v, (z,t) 1 e 2 01 (3.28) 2j (6 -5 ) ( -6 ) 1 2 1 3 which, when multiplied by the input with frequency cw, gives the first 2 approximation to the circuit voltage with frequency c. 2 Second-Order Solution The second-order approximation is obtained from the general Eqs. 3.16 and 3.17 by letting s = 2, w = O; s = O, w = 2 and s = 1, w = 1. Considering the case s = 2, w = O, therefore m = O, n = 6 or 2. For s = 2, w = O, m = O, n = O, Eq. 3.17 gives _(2,0) c+j o) 3F?p (6,p) = P(10) ap B C 6 2itj 1 [- 1 10 10 c- jo _(1,o) (1,0o) ~ F (r-p) [ clo(5-r)-jp ] + V (5-r) 10 10 10 -10 10 (F1)0 (r,p) [ lOC (-r) - j ]dr -1.70 10 CO 10 (1,0) _(1,) (1,0) (1,0) Note that V and F are obtained from V and F -l0 _-10 l~O lYO respectively, by changing j into (-j). The complex integration on the right-hand side is carried out only over the pole with the largest positive real part. Carrying out the complex integration and then integrating with respect to p, the result is:

-23(2)0) 2 -pu F (6,p) = a eo 4 jC3 u45 10 0 6[ 1)2(Co (55 61) + j[(PUo(-j +PUoC1o01) + (j+2pUoC 6 ) 2C 6 i IO 1 0 0 10 1 010 1 10 1 ( -6 ) ( -6 )[(6-6 )2j (-6 ) + b} + 1] 1 2 1 3 i 1 (656*)2 (C (5-6*) - jpuo(j+puC 0 ) (j + 2pu(C + + 2C 1 110 1 10 (6*-6*) (8*-6*) [(6-6 )2{- j(66_ ) + b} + 1] 1 2 13 1 1 (3.29) For s = 2, w = 0, m = 0, n = 2, Eq. 3.17 gives (2,o) P (6,p) 2jcD (2,0) c+joo 2,o - p) 1 20 ~ + -1 - F (6,p).. 6p jp p c 5 2 0 jp -3 PC 6 2xj. 10 10 10 1~,o 1'o - 0- ZO JD2o-f1oClo c-_ -1 J zoo(2,0r) F (r,p) (2,0) _( 0o0) - + [10oclo(s-r) -j V (6-r) (-r) dr. Carrying out the complex integration on the right-hand side as before and then integrating the resulting linear differential equation in _(2,0) 2,0 (6,p) gives

-240 i~~~~~~~~~~~I —I1 0 Cl) 0,.0~~~0 oX)~~ I ~CO~ c -C rl0 0 cocu 0u~ CH~~0 H 4' + 0 co ellr 0 l r3 C\j CM 0 -. 0 0 ~\ r ~~~~~~Cll 0/~ 0 co c 0 ~ ~ ~ ~ 00 ~/I O: (V i-1 to 0 H I:,-II \I 0 CU U 0 rl 0 ~0 3H H 3-:~ Ci)' — ~r C ~ ~~~~~U ~O H-I 00 0 * ~ 0 ~0.5 I ~ I, - - ~ ~ --' e —- 1 ri 0r4~ ~ 0a)t0 o c Hc~ to + 0 I 4'o H +o o l-rcu i a o 0 Oa co~~~~~~ HU co o~ Cll ~ 0 v CO + 4 ~1 d rl ( i.O c O, —\ +0 C0 c' O 0 o oI ~0 coi 0. 1 co 0'I H ) ) H I.0 0 0 cm 0 M C c Q CC ~0 co co.0 Hq 0 1 - N I O Cc C o ) 0I H_ P4 co % -j co c H H ~ CUCU i OI 0CM - 00 C. U) H I 0 r..3 II CM N LO~~) 3,1* II 0 0 0 0 ~0 o I I ~0 H 0 I 0> ~ ~ ~ ~ ~ ~ ~ C 0L'D,.4 ~0 H-.. 0 ~0 ~0 0 Cl U) 0 co +. H o j r H I,~i — HorH-F-4 H I4 P-1 C\ l b r o~~0 CU0Cx o co r~-i cor- ~0 Ci C 0 0 to 0' H. CU H I I C CM ~ 0 H~ C 0HC) CM +.rl C 0 ~ U0 C) 0 C 0 11 p C ~ P0C 0 I ~0 I0) * 0~ - M U' crCU -') CM 0 C Cl H C O Cl OH CU CU ~ ~ ~ 0 CU 0U U) a IU~~ ~ H Ii PI 4 CU Fe b) 3 ~0~~~~ H ~0 C)4-. —-4) 4_ I -~~~~~~~~~~~~~~~~~~ 4'~ ~ ~~~~~:

The pole with the largest positive real part of Eq. 3.32 is 5 = 2 5. Therefore, (2,0) V (cp) 2,0 62 ~,C,, 2j(w t-~ z) 62 e2 10 c 10 1 10 3 C2>o3 2 01 e = +1 (-, C3 1 0 v a2 (6 -6 )2(6 -6 )2 20 0C1 ~ 20 + 20 0 01 1 2 1 C3 1L 10 (3.33' where C20 in the gain parameter at frequency 2X. If the circuit is nondispersive then b20 will be equal to b10, and therefore 62 (-jb+b ) = - 1. If (C /Cl )3 can be neglected 1. 20 20 10 compared to unity (which is almost true, since the circuit impedance at the harmonic frequency is much smaller than its value at the fundamental), Eq. 3-33 reduces to the following expression: 2 C 6 2j(wo t-$ z) 2,o) C_\ 2e e (2,o> - 1-Y Clo/o~3 2 v2 0 (cp) = 7 v 334 102, (V a _ )(6 -) 0 10 12 13 Multiplying Eq. 3.345 by B2 and taking twice the real part gives the second harmonic voltage of the signal at frequency c1. It should be noted that the above expression gives the first-order approximation to the second harmonic voltage. Higher-order approximations could be obtained. However, the next order will be proportional to B4 and to 1 B2 B2. This will not be carried out here. 1 2 By the same procedure, the second harmonic voltage of the signal at X can be obtained. The result is 2

-262p 1C 5 2j(co2t- olZ) (0,2) c 3 62 e (cp>~3 o 02 1 0i2 jJ Vo0 v0c(6b 162)2(6 )2 Third-Order Solution The third-order approximation to the voltage at frequency u is 1 obtained by putting s = 3, w = O, n = 1 and s = 1, n = 1, w = 2 m = 0. The first case gives the dependence on B as B3, whereas the second 1 1 case gives dependence on B and B as B B2. 1 2 1 2 _(3,o) aF0o( 5(,p) ~1 (3) o + ~~ (6,p) = - 1ap 10 1,0' 10 10 0o c+joo (1,0) (2,0) 2 j.. LV (5-r) F) O (r,p) [5 c.o (-r)-jp ] 27c.j L 1.~ 0,0 10 10 10 c-joo (1,0) _(2,0) + V 1 (6-r) F2 o (r,p) [loclo (6-r) - j_ 1 o (2,0) _(1,o) + V2 (6-r) Fo (r,p) [~c (6-r) - Oa 2, -1,0 lO 10 (3,o) _(o,o) d + V1 (6-r) Fo (r,p) [ 0loClo(b6-r) j- 10] dr (336) To facilitate the calculations, the second harmonic voltage will be neglected. Use Eqs. 3.23 3.27, 3.29 and 3.30 and integrate over the pole with the largest positive real part, then integrating the (3,0) resulting first-order equation in F (6,p), the result is 110

ol R 0- %'ro 0. % - o o 6 a ol ~o,C? h oo PS L o o 0 o 0~~~ o Oo 11, ta 0> ~ 00, ol @00 c r~~- 00, o, co, 0- to & Y L~~~~~o ol ~ ~ ~ ~ l _s 0 O 00, e0 0 * ro oh P&* oO ~ P_ o o o Fi ~ s" b oo _ N +N O vQ~~~H o ~~~~~~~co 00{ r *I 00. cs "*~~~P r (P C~~~~~@& O_ r r" r.d, h)~

-28where A = u C1oA/(j - C 6). From the circuit equation (Eq. 3.16) for s = 3, n = 1, w = 0, (3,o) (3,0) V () [2 +V2 2C2 52 2j2 c 6-2)] = - z v o2 0 (6,0) v,~ () [1 + 0 l o10 10 o o10 10 10 1,o ( ~~~~3, o) (3.38) By solving Eqs. 3. 37 and 3 38 for V1 0 (6), the result is 1~0

-29cu o 0 C) o *CH Cr.) v CO + M co CU' Ho CO C,o0 c F *o cO o 0 I *H H 0o 600[. to Io "" 6 COH 0 Ho c o C o -I.-I O C V + o to, -I C,-I C c O0 ~" CO H\ o I CU 0 60 N H H' C o+ N H 6 C\ C0 C 110 60,46 H I CM t0 + OCH CM ~O h~0 C I\> IH 6

-50By taking the inverse transform of Eq. 5.59, neglecting terms of order of C compared to unity and rearranging terms, the final result is: (3,0) V10 (z,t) J *5X2 *2*2 * 1oC1 (26 1 K +)z j(cow t-S0z) [51 - +36 +56-(31/2) ( 26 4+)] e e 32V2C4( ~- )2(6 1-6)2(6*(1-62) ~1*1321)( 1'(1+~1 1 (. ) Equation 3.40 does not give the total third-order approximation, it is only that part which depends on B1. The other part depends on B1 Bz, it (1,2) is V1 0 (5). This is obtained from Eqs. 3.16 and 3.17 for s = 1, w = 2, m = 0, n = 1. Equation 3.17 gives (1,2) aFlo (5,p) j (12) op jppju c C 1,0 jp -p c6 2 j 10 10 10 10 10 10 c- joo r(o,1) =(11) L Vo (6-r) F11 (r,p)[loCo(-r)-j ] (0,1) _(1,1) + V (6-r) P (r,p) [ loc(-)-j O (,o) _(0o,2) + V1 0 (6-r) PF (r,P) [l~Cjo(5-r)-jo ] ( 1 O,) +V( )(5-r) F (r,p) (1,1) =(0,1) + V( i (6-r) Fo (r,p) [1clo( -r)-j1 (1,2) _(o,o) + V1,0 (6-r) Fo0 (r,p) [Clo(-r) - ]dr (3.41)

-31As before the second harmonic voltages will be neglected, also the voltages at frequeiicies l + w2 and wl - w2 will be neglected. These are reasonable assumptions when the frequenicies c and. o are not 1 2 very much different and both near the center of the band. It should be noted that although V (6), V (6) can be neglected but F (6,p), F (6,p) cannot. This is because the harmonic content of the oeam might be large, but its effect on the circuit is small because the impedance of the circuit at these frequencies is small. To evaluate Eq. 5.41 one needs the following quantities in additiontothose previously derived: Fl 1 (6,p), F1 (6,p) and (0,2) 00 (56,p). Using the same procedure as before, these are: O,O

-52* - - to Oj H _ O 0 I C\' cO ~0 Ho0 I O 0 0 I0 0O 0 ~0 CU 0ab oo o t o H m 0 O V O + c~. H H c 0 0 0. o H0 H ON 0 cO, — I rOH H H * H 0 co',1 ~0 0 (-. k 0~H 0 0 O 0! ~0O co H;0 N r~,-H 00 0 H$'r1H H 0 I~0 ~0_3~ ~ ~r 0 0 Co 0, -0 H.. H co o 0 ~0 I + I 0 +\ ~0 C. I —,, c:o d~ ~ ~O o L? LO 0 cU [V 0 COI I 0 cO * CO H H H - 0 H ~0 c0 o0 r00 0 H 0 CO 0 rH ~0 I L H o 0 II ~ ~ ~ ~ ~ ~ Or

-33o CO cO LH I O H C 01 0 H C Soco H H 01 4C U 0O 0 H H Io I O 0 0 H1 0 c0 + Co_- O 00 00 O H) +oH o v v0 H U ) X H'O ct ~~~0 HH 0 \ O O C) 0 c Ho \O H 0 CM) CCO H -0 CMO L' H CO f 1 CO 0 H 0 C O 0 0 H CM CO 0 0 HO C 010.-0 r0 M\j~O 0 cr. H L J H CO 0 0cx>~~~~ Ho 0 co0 H co c0C O OHO C H CO 01 CO 0 HO 0 O H 01 CO 0 H 01 0 H I U0 HI 1, I-I I r i~~~~~~~~~~~~

I cZ I v 60, + o Ho,6 r Vr ~ ~ c ro 1 H 0r co o. o 0 + + H 0 V o H i C H 0 0 / + 0 H O O tO r.O o 0 I O H 0 0 60 0 0 OV I IH O 0 OO H-i H + rOi H 0O 0o i 0 0 0t tO o r 0 +0 +0 H 0 C a oL Ca GI V 0 0. r C Dr_ l I." ~O _, 0co 0 rl o O o ~ I CO (33 rO r0 C 0 - I HI H 0O N iY) f-i r-^ O i 0 0~60 H H to I0 r CI) co rli COH H H.0r-a 0 0 C- i O 0- /0 Ho O OJ1 0' - O * H 0.H CU:J rOrH0 C 00 H C( * ) Co tzf O rl CO ~ I OI{ I C.. 0 I - 0 cO O O ~e~i+

Using Eqs. 3.42, 3.43, 3.44 and the previous quantities, it is possible to evaluate the complex integration of Eq. 3.40 and then solve the _(1,2) resulting linear first-order differential equation in F 1 (6,p). After _( 1,2) ( 1 2) 1, obtaining F1 0 (6,p), together with V (5) (from the circuit equation), (i,2) 1,0 one can solve for V (b); the result after taking the inverse 1transfor transform is

C) C0 -36-~~~~~~~~~~~t,-I ~~~~~0 H 0 0 LO -4~~ -! co t o cO rO -Fco o~ o ~o ~ ~~~~~~~~ o 0 0 ~~~~~~~~~~~~~~~~~~o CEL~~~~~~~~~~~~~~~~~C c~ cO~~~~~~~~~~~~G 0 co- co-0 c 0 + ~ ~ ~ ~ ~ ~ rl U ~ ~ ~ ~~~~~~~-I L J HL 0 O ~~~~~o H - C) HLO 0 — 00 Ca H I 0 dl 0 0 I HH 0 3-~~~~~co C0 C) 0 + Ca co Ha 0 0C o H + o H 0 C) C)0 u 0ca 0 \i co ~c 0 Ca ~I C) oo -I CMH o 0O 0Ho H N0~~~~~ --- 0 cO Ca C)Ca Ca-I 4c Oa ~ ~ ~ 0ci 60 +) H..H L1 H H C) C) Ca ~~~Ca H 0 H- n NII cu co, - I C + H co Hn CaOIC) Ca H0 1 r-iN00 HoC) CM C0 0 0C 0 0 o 0 to O — I N 0 4.~ 0 Ca a H Ca~~~ ~ ~~~~~~~~~~ CaJ ~ 0t + 0l ~ H 0 C) C) H H 0 CM CMO -~~.C.............an Ca *C Hco HZ C)E C) C 0 H H + Ho 0 0 H 0 H 0 H Ca~~~0 c * u H H l-O —-J 0 + H C) Ca Ca C)0 C)- Ca - O c Ca Ca~~~~C H H + +0..~ r. c:,:-IH +co~0 0 H tO C) H —I C c,-t CMO- Ca)- i + C) C0 HH O >k r, 01 r co r-i r-i * H~~~~~~~ r CC) r-i~~ 0 C C (0 +1 CaC 0H 0 ~ 0 c~~o co0.iu1i1 I o Sl N Ca~~~~ * 0 L + CIO Cc LO CU~~~~~~~~~~~~~~~~~~~~~~~~C r-i ri CO C0 0 _ c) C) H) I 0 HoH + 0 0 10 U + UH coL C Ca oI) Ca 0 + N 0 +4 + H > C 0 -0 0 61 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _~ L' \/ 0 rl Ic- o H C) H H * H + ~~00 (0 0 C) + Ca CMH H CM 0~~ ~ 0 0 06 - C)C ). Ca c CM HL H1 0 C) CM C) C'o- C 02.,-I r, sO 0' O CX) * H C) Clrl M 01 H 0 0 H aC O U U LO ~~~~~O O~ I~ CaC Hl +l H c f cO O O re +,,, ~~p 4rH C) 4~~~) H( C 0 C) C) cr1 CO. C)~~L C) HU + +l + C)~O r(1 O ~ er

-37Multiply Eq. 3.40 by B and Eq. 3.45 by B B2 and add; then take 1 2 twice the real part, which gives the third-order voltage. In this analysis it will be considered sufficient to stop at the third-order approximation. High-Order Difference Frequency Components If the frequency difference (wo -o ) between the two input signals 1 2 is not large, according to the general expressions, Eqs. 3.5 and 3.6, there will be components with frequencies in the passband of the circuit. Two of these components are with frequencies ao -w and ao -o. The 1 2 2 1 component with frequency a2 -w will be derived. 1 2 Substituting s = 2, n = 2, w = 1, m = - 1 in Eqs. 3.16 and 3.17 one gets: (2,1) V2, (~) [(a.-w) 2+v2 _1(p2 02 52 _ C 2)] 2-, 2 2;~ 10 10 10C, 2~- (2, ) _(2,1) aF (6,p-1 ) + j( 22) F(2-1 +,(o F (6,p) ~ o- or(2,i) (o, o) - f Lo(b-r)-l2,-.z (6-r) F (r,p) j(2P -~o~) -~ c 5 2-j (1,o ) (1,.) (B C (6-r)-jp 3 + (6-r) F (.r P P) 3.03.0 3.0 v3.,0 1(-r) F (r,p) (0,3) _(2,0) 3.00loC(6-r)-o) + Vo 0,-i (6-r) F(2,0) (r,p) oC0o(5-r)-J o,_] 0 dr (3.47)

-38Integrate the right-hand side and then solve the linear first_( 2 1) _(2,1) order equation in F2 (6,p). The solution of Eq. 5.47 for F (,p) (2,1) y is solved simultaneously with Eq. 3.46 for V2 (b), and the result after taking the inverse Laplace transform is:

-39co * H ~0 H CLI ca.1 o~~~~~~~~~ H coo O H 00 K0N v H * H 0 ~O 0 0 0 01 O cci co. 0 CC r-iu to -) H 0 C~~~~~~~~~~~~~~T~ 0o o H CO H GO ap~~c H U U-r *O UL O. 0 0 to 0 H 0 H H U c0 0) o + ~~~~~~~~~~~~~~~~~H + o 0 01 0I H -% 01 1 HO H * % OH 0 0 H rl r U H 0 O U + (OH 0 H~~~~~~~~~~~~~~~~~~~~~t H 01 H0 0 to H H 0 H t0 01 0 0 C O o. 0. H0to 0 H ~ ( I' H 0U 0 0 ~0 o H o ~d H Cv O 0.O co. 0 -- I co_.Oar C\ r CO + to~3 c (O 0~ 0 O @1 H 0I C U.0 0o H o 00 H o OC + 4-' 0 ~0 to~~~~~~~~~~~~~~~' o ~4 0 U OH LO CO Hl Hr 0 3 1 0 0L U ~0O U 01 ~ I H H'~~, CoO ~ 40 N to r-i H 0 CO H ~0 H c4 * H U 0 0?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- totoc C co 0H H 0o 0 H 0 rl crO r(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- n -__. N 1C *- - H ~ Uo'H? ~00 rl HO 0 co 00 H co r0 0 0 0 C + o 0 U I H Ha O H ~ 01 0 M 0 I 01 C H ~0 H ~ 14 O \f-,l r~~~~~~~~~~~~~~~~~co- a 0- U H 0 U 00 0o O01 - C *. 0C O0O~~ H + OHDr- - CO H U CO H ~~~~~~~~~~ ~~~~~~~~~~~~0 * 4 H H 0 co ~01 013 ~0 0 H0 LO co O H 0 C\ o 0 _c_ O 201 0 o0 + U I IH 0 ~0 4,O O 0 H C\O0 H CO COH H-4 C\ ~0 0101 0)- CM. + ~0r t ~0i t0 C\l ~ ~ ~ Or + - H 0 H H COH CO0 ~H 0, H —----- o o CO o 0 + 01He o C dL 6 ~~1 * N~~ 9l r- - r. D I ~0 H to 4 HI o Ti HI~ H~ ~ U UH 0 1 0 * H 0 0 H 0 Hc0oo HO 0 H\ C0+ -0 U~~ UU 0 -O H0 O 01 01 011 0 o~co C)0 H_~ 0 r-4 rlO 0 +H H H U + CO H ~0 U 0 ~0 ~~ ~ ~~~~~~~~~0 0 0cH ca cU~~~ 01 COO * H 0 H C, 0 Ho 0 co Ho 01 01-ir CQ. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~~~~~~~~~~~~~~~.. ~ ~ ~~~~ I 0~~~~~~~~~~~0~~~o CU~~ ~ ~~ HO'r I N 0 14 ~~~~~0H0 II 1~~~~~~~~~~~~~~i 01 II H (U ~ ~ ~ L I 01 3 60-~y-l~ f f ai, ~0 CO CM C>~~~~ ~ ~~~~~~ + i O

-40The circuit voltage at a2-W1 can be obtained from Eq.- 3.48 by interchanging 1 and 0. It should be noted that the circuit voltage is obtained from Eq. 5.48 by multiplying the right-hand side by B2 B2 and taking twice the real part. Other intermodulation components can be obtained in the same way. 3.4 Program for the Next Quarter. The equations presented in the first quarterly progress report are being programmed for digital computer solution. The results will be presented in the form of curves representing different operating conditions. Also, it is intended to program the equations of the second progress report. No experimental work has been done during this period. However, it is planned to initiate experimental studies on an X-band TWA during the next period. 4. Study of a D-c Pumped Quadrupole Amplifier (C. Yeh, B. Ho) 4.1 Introduction. In Quarterly Progress Report No. 2, the state-of-the-art of cyclotron-wave devices was reviewed. It was suggested then that a unified analysis should be carried out so that the coupling mechanism among different modes of operation could be better understood, and a fair comparison between the different coupling mechanisms could be made. To achieve these objectives, an analysis based upon coupled-mode theory was carried out during this reporting period. Although the coupled-mode method does not represent an accurate picture of this device, especially for large-signal operation, the simple mathematics it uses enables one to visualize the physical insight of the device and, in particular, the interaction mechanism between the different modes of coupling.

-414.2 Coupled-Mode Equations of Beam Dynamics. Consider a filamentary electron beam in spatially varying electric and magnetic fields, the transverse equations of motion in rectangular coordinates are d dtVx = (Ex + vy - v ) (4.1) x x y z z y and d - v ( E - v-B +v B) (4.2) d-ty y x z z x where q is the charge-to-mass ratio. The transverse displacements and transverse velocities are related by dx d = v (4.3) dt x and dy = v. (4.4) dt y With a static axial magnetic field B = B., and assume constant z 0 = u, Eqs. 4.1-4.4 can be put into a set of coupled-mode equations z 0 in terms of mode amplitudes as follows: (d +j) a = - k(E + joB) + a d, (4.5) (k-+ jw) a = - k(E juB) + d a (4.6) dt c 2 o- kdt 2 da dcu 3 1 dk 1 c dt T - k(E + j u B) + - dt a + 1 (a -a), (4.7) dt = k(E - j UO B+) + k dt3 w+ dt 3 1 da 1 dk 1 dLct dt - -1 k - Uo - o kd-t 4 dt 4 2 c

-42where a = fast cyclotron wave 1 _j( - )z A e k(vx + j Vy) (49) a = slow cyclotron wave 2 -j(13 + c )z A e 2 k(vx - i ), (4.10) a = negative (kinetic power) 3 synchronous wave -_ z =A e 3 k[vx+j vy- jic(x + jy)], (4.11) a = positive (kinetic power) 4 synchronous wave -ji z A e 4 k[vx-J vy+ jc (x - jy)] (4.12) A. = amplitude of the wave, i = 1, 2, 3, 4, 1 k = rlu// ) = -' - c e u'c u o o E+ = E + j E + x y B+ = B + j B x y co = B B c o The transverse displacements and velocities can also be expressed in terms of mode amplitudes by using Eqs. 4.9-4.12, as

-43 - a -a -a +a 4 2 3 1 (4.13) 2 j kw a -a + a - a 4 2 3, (4.14) 2 kw a + a v 1 2 (4.15) x 2k and a - a 2 4.16) y 2jk (4.16) 4.3 Coupled-Mode Analysis of D-c Pumped Quadrupole Amplifiers 4.3.1 General Procedure of the Analysis. To simplify the analysis of a d-c pumped quadrupole amplifier, let us assume that the a-c magnetic fields in the transverse direction are negligible, i.e., B+ = 0 and that c is a constant and that dk/dt = 0. The coupledmode equations (Eqs. 4.5-4.8) reduce to d (dt -i j) a = - k E (k+j wc) a = fk E da 3 k dt = kE da 4t = k E. (4.17) dt In order that parametric pumping can be achieved in a d-c pumped quadrupole amplifier, the pump field must be an appropriate function of x, y and z. Let the transverse fields E and E be a function of the coordinates of the form,

-44E = f (x,y) f (z) + 1 2 E = f (x,y) f (z). (4.18) 3 4 By means of Eqs. 4.13 and 4.14, it is possible to express E and E as a function of the coupled modes a -—, a. Thus, one may write 1 4 E =f (a,a,a,a ) f(z) + 1+ 1 2 3 4 2 E = f (a, a, a, a ) f (z). (4.19) E1- (-1 2 3 4 4 Hence Eq. 4.17 can be rewritten as d ( _ j t ) a = - k f (a, a, a, a ) f (z) dt c 1+ 1 2 3 4 2 da 3t = k f+ (a, a a ) f z) dt tc 2! 1 2 3 4 4 da dt - k f (a, a, a, a ) f (z),(420) 1- 1 2 3 4 4 Equations 4.20 enable one to discover which pair of modes has strong coupling. Depending upon the actual field configuration, the f and f may or may not contain all the terms in a's. By inspecting 1+ 1each part of Eq. 4.20 separately, one may identify which of the pairs of modes are coupled together. This point will be made clearer in an example in a later section.

45 Equation 4.17 can further be simplified for the case of constant axial velocity and a d-c pumping, i.e., v = u constant and w = 0. z o Assume sinusoidal variations, then d a a i j + u z o az o az Equation 4.20 becomes j )al= - a E, (4.21a) (az 2 u - 0 oa - E, (4.21c) )Z = - u + oa 4 - l - E (4.2ld) aZ U In order to examine the effectiveness of the mode coupling, a set of second-order coupled-mode equations will be derived. Differentiating Eqs. 4.21a and 4.21d, subtract, and then adding Eq. 4.21d which is multiplied by j c, one obtains after some simplifications, the following expression: jc z 82A aA 62A a A e ( b z2 + J 3c b~ —z2Caz u + Similarly from Eqs. 4.21b and 4.21c, the following is obtained: -j z 82A aA a2A aA e -a2 a-Z - a2 -. = i E. (4.23) 2Equations c4.22z and 423 arez cthe general expressions which can beu used to analyze all types of d-c pumped amplifiers as long as the pump field E+ can be determined.

-464.3.2 Staggered Quadrupole Amplifier. As a typical example, a detailed discussion based upon Eqs. 4.20, 4.22 and 4.23 will be given for a staggered quadrupole amplifier. The pump electrode configuration of such a device is shown in Fig. 4.1. The potential in the pump region is given by V(r,, z) = V ( )2 cos 20 cos 0 z. (4.24) p a q In rectangular coordinates, the field intensities are V E = - 2 P x cos t t x 2 C a E = 2 P y cos t,(4.25) where Pq = 2i/L. Then 2V E i (a - a ) cos B z + aa2 k 4 2 q 2V E P (a -a ) cos 3 z (4.26) 2k, 1 3 q Substituting these quantities into Eqs. 4.20, one obtains d 2" V dt - j ) = - j (a - a ) cos qz, (4.27a) d i ca2 = 4 2 q c d 2j V + j a P (a - a ) cos f z. (4.27b) c 2 2 1 3 q a O c da 2i V (a- j. (a - a ) cos 5 z, (4.27c) dt 2 4 2 q c dt4 - -. P (a -a ) cos z,(4.27d) d_t a 2 1 3 q a C2

-47 - -Vp a -VP p VPP I -Vpl Vpl I FIG. 4.1 GEOMETRICAL CONFIGURATIONS OF A STAGGERED QUADRUPOLE PUMP STRUCTURE.

48 The important feature of the parametric coupling can readily be seen from this set of equations. With a staggered quadrupole d-c pump field, the following mode couplings are possible, Eq. 4.27a indicates the possible coupling between the fast cyclotron wave a to 1 the slow cyclotron wave a and the positive synchronous wave a, similar 2 4 coupling pairs can be sorted out easily from the other equations. However, two types of coupling are not possible, namely, a and a, and 1 3 a and a 2 4 Let us proceed to study the different cases of couplings more carefully. Substituting the fields E+ and E for the staggered quadrupole of Eq. 4.26 into Eqs. 4.22 and 4.23, one obtains 62A 6A i z 2A 6A jp z -jBz K +j 1) e\ c + j= M Ae +A e z2 c z2 c 4 j(13 - A )z -j(e + z - A e c - A e q e (4.28) 2 2 and a2A aA - j z a2A aA Z -j z j~,...2 -Jc 4j qj 4q (~i2 jpc 2 )e - - j =a M [Ae q + A e Z C z 6Zb2 C Z3 3 j(q+ ~c)z -j(q - 0c)z - A e q c A e ], (4.29) 1 1 j where M = (VpV ~c/UoWca2) = (Vp/2Voa2),and the A's are the amplitudes of the coupled modes. Equations 4.28 and 4.29 may be used to discuss the coupling modes and the gain of the device. There are several cases of interest

depending upon the spacing between the quadrupole sections, which can be discussed separately. Case I. Coupling between Cyclotron Waves. Let the spacing L = Xc/2 or P = 2 P Equations 4.28 and 4.29 become La2A aA z 2A A j2 z -j23 z Le+ j.... = MA e + MA e 2 c a2~2 a4 4 ji z -j3pCz - MA e - MA e, (4.30) 2 2 62A aA -ji z a2A aA j2 cz -2j~ z F___ - 4. 4 c 2-j ----- Je -4 j - = MA e + MA e Lz2 caz az2 az 3 3 j5 z -jP z -MA. -MA e e (4.31) 1. 1 Equating z-dependence teris, a2A aA 1+jB _- j -MA, (4.32a) 62A aA 2 2 2 j A = -MA, (4.52b) az2 c az 1 c2A aA (4.32c) az2 az 82A aA + jp ~ = 0. (4.j2d) az2 c 8z

It is obvious that there is no coupling for the synchronous waves, but the fast and slow cyclotron waves are closely coupled. The roots of Eqs. 4.32a and 4.32b are = -1 + 41 + 4 (M/12)2 (4.33a) = j +J1 +4 (/) = (4.33b) 2 (/c 2 1 +N41 +.4 (M/2) = (4.33c) 4 C + + / + 4 (M/2c = j2 (4.33d) where p 1 + 41J + 4 (M/p2)2 42 It is noticed that y is a positive real quantity for all possible M's 1 (M varies as V varies) and y is pure imaginary. 7 = - y and 7 = - y. It is clear that a growing wave is possible in this case. 4 2 The solutions for the fast cyclotron wave a and its slow cyclotron wave a are, respectively, 2 jp z a A cz a = A e 1 1 7 z j z -' z -j z jf z = (f e 1 + f e 2 + f e 1 + f e 2 ) e (4.34) 11 12 13 14

-51and jp z c a = A e 2 2 y z jg z -Y z -jf z -jp z (f e f + 2 1 +f e + f e (4.35) 21 22 23 24 It is noticed that there are four component waves in each mode; one is a growing wave, one is a decaying wave and the other two waves are constant amplitude waves with phase velocities corresponding to +~ c 2 c respectively. The phase constants of these component waves are plotted in Fig. 4.2 as a function of the pumping parameter M. The amplitude gain of the growing wave vs. pumping parameter M is shown in Fig. 4.3. Case II. Coupling between Synchronous Waves. Let the spacing L -, oo (i.e., single extended section). 2t = 0 q L From Eqs. 4.28 and 4.29, one obtains a2A aA jAz 62A zA LZ2 + j ec e 3 + jc 3 -j~ z - 2M (A -A e ) (4.36) 4 2 and 2AA -jz a2A 1A jg z j2 e - 4 j. 4= 2M (A - A e ). (4.37) 5z2 c 5z 2 c 5z 3 1 Thus, 82A aA + jB _ = o, (4.38a) a~2 aZ

-520 0 E-i o o oo H 22 H 0 o ~~~~00 CM 0 0 >~~~~~~~ o m p Cu a.~~~~~~~~~I a. EJ~~~~~ M~~~~'i~~~~~~~~W w 14 PRD~u P E'4- ~ ~ ~ ~ ~ Q'4- Ii 0~~~~~~~~~~~~0 z~~~ or 0~ ~~~~ > o 0 ~o~ ~~~F:-~~~~~~~~ S':IA~/tA'.LN:3NOdlIO3 3H.L 10-,1.iOLN~.LSNO3:\SVHd

o o CCn S H 0 EN r H H \ UV CNH \N ~Q~ uD e- i~ 0NO~~~~~~~r

32A aA )z2 c az 82A aA 3 _j 3 - 2MA, (4.38c) 6z 2 c 5z 4 62A aA 4+ j _ 4 - 2MA (4.38d) aZ2 c az 3 It can be seen that there is no coupling for the fast and slow cyclotron waves, while there is coupling between the synchronous waves. The propagation constants are found to be C 1 +~41 + (4M/)2, (4.39a) = - - 1 + 41 + (4M/i ) - - (4.39b) 2 1 y = C + i1 + (4M/2)2 j= (4.39c) j - l~4l+(4M/j2) = lj 71 +/41 +4M/2)2 = 4- (4-39c) They are of the similar form as in the previous case and thus gain can be obtained in this case. Case III. Coupling between Cyclotron and Synchronous Waves. Let the spacing L be equal to the cyclotron wavelength I; then, =f f. ( Equations 4.28 and 4.29 become q c

2A A +j3 z a2A aA + 1 + c 23 + j a 3 = M e + A e -A -A e (4.40) 4 4 2 2 and 62A z A - — ji a 62A 3A La22 j a 2 Ie c 4 j 4 c 5z 6z2 c z jp z - j z j2pz = MA e +A e C - A e - A A (4.41) Equating the corresponding z-dependence terms, one obtains 62A aA 1 + j 1 = MA, (4.42a) aZ2 caz 4 62A aA 4_ j( 4 = MA (4.42b) az2 C z 1 a2A aA 2 _ j 2 = MA (4.42c) 3a2 AaZ 3 32A aA 3 j 3 =MA. (4.42d) 1. Coupling between fast cyclotron wave and positive kinetic power synchronous wave. The coupling between A and A can be analyzed in the following 1 4 manner. By solving simultaneously Eqs. 4.42a and 4.42b, one finds the roots of y of the characteristic equation ( 2 +..) - M2 O as

rY i2 (1 - - J4m/2)) (4.4a) = 2 (1 + i- (4M/2)), (4.4b) 2 - =y i 2(4M/C) / ( 4.45c) 3 2 c y = - * fT4/2 (4.43d) 7-v- (1+ 4 J There are two possible cases of operation: a. For M < p2/4, all four y's are pure imaginary, the amplitudes of the component waves are constant. A and A are said to be coupled 1 4 passively. b. For M > @2/4, y has a positive real part, which indicates c 1 that f and f are growing waves, therefore, gain can be obtained. 11 41 The o-B diagram for both cases are shown in Figs. 4.4 and 4.5. The phase constant for the component waves as a function of pumping parameter M is shown in Fig. 4.6. Notice that for M > 2c/4 the component waves f, f, have constant - = -c/2, and have growing and decaying 11 12 amplitudes. 2. Coupling between slow cyclotron wave and negative (kinetic power) synchronous wave. Consider Eqs. 4.42c and 4.42d, the propagation constants are found to be 1 = ji2 (1 -%!i- (4M/@2) ) (4.44a) 2c (1 +4 —(4M/@2)), (4.44b) 2 2

-57N u-~~~~~~~~~~~~~~~~~~~~~+ 0 z H q~~~j. ~~~H CY~~~~~~~~~~~~~~~~~~C E- r 0 0 I P rlaz~o H Cl) Q X E 0 <I H 0 ~I re v, U3~~~~~~~~~~~~~~+ r Q o e;~~~~~~~~~~~~~~~~~~~~~~~. d`~~~~CL

QV H o 0 E\ cQ \ 8 | |Q~~~I H o \ E —i I 0

-59-.J0~~ I'" o "" 0 H H' 0 a.a ~~u ~ I o 0 o m w w~u >9 wE-i. 0 a. o~~~~~~~~~~~~~~~o aw w < 0 ~ 0 I.UZC H I 0> -i —------ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~E c~~~~~~~~~N 0 0 f\~ OD 0 o \1 (, o I6~ ~ ~ ~ ~ ~ ~~~ c i p I wZ~I COD~~~~~~~~~~~~I o Ci A.. 0 ~~n E(~ 0 0 0 0 0 0 0 0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 N N (0 CD N~~~~~~ o a 0 6 a a I H S3IWM 1N3NOdbr\JO3 3H.L tJOd ~N~SNOZ3 3SVHd -

-6oY =i 2 (1 - 41+( 4M/2)), (4.44c) C C y = 2 (1 + 41+ (4M/2)). (4.44d) Again there are two possibii.cies as in (1), when 4M/1c2 > 1 gain is observed. 4.3.3 Other Pumping Fields. 4.3.3a Twisted Quadrupole Pump Structure. Figure 4.7a shows the geometrical configuration of a twisted quadrupole pump structure. The equation of the potential in rectangular coordinate system is 1 p [(x2 - y2) sin 2pz - 2xy cos 2pqz]. (4.45) 22 The equations of the polarized field in terms of the coupled modes are V j2e z E += - P (a -a ) e q a2 4 2 c V -j2P z E- = - P (a -a ) e (4.46) k a2 3 1 c The propagation constants for coupling between the fast and slow cyclotron waves for pq = P coupling between A and A q c 1 2 are

Vo qx a) TWISTED QUADRUPOLE STRUCTURE ~~y ~-Xq-I a R -VR VR -VR b) PERIODIC RING STRUCTURE x Vp -Vp Vp a a ~_ _ - - - z nnn c) ELECTROSTATIC SLOT STRUCTURE FIG. 4.7 GEOMETRICAL CONFIGURATIONS OF (a) TWISTED QUADRUPOLETYPE, (b) PERIODIC RING-TYPE ARND (c) PERIODIC SLOTTYPE OF QUKDRUPOLE PUMP STRUCTURES.

-62YC 2 1 - + - $ ~+/l+4 (M/ )2 - + 7 c 2 3~3 q 2 - + C0 c2 - + j4 (447) 4 For the coupling between synchronous waves A and A, = 0, and X = 00. Roots of the propagation constant y are the same as in the previous case, indicating the possibility of obtaining gain. For the coupling between cyclotron and synchronous waves, 1 Cq 2 c'q c For the coupling between A and A, 1 4 y _j / c(1 +24l-( 4M/2)) 7 2 c 2 4 It can be seen from this equation that the passive coupling occurs for 4M/p2 < 1, and the conditional coupling occurs for 4M/" > 1. C c For the coupling between A and A, the same criterion applies. 2 3 4.3. b Periodic Ring Quadrupole Structure. Figure 4.7b shows the geometrical configuration of a periodic ring-type quadrupole pump structure.

The equation of the potential is 1 p (x2 + y2) sin X z (4.49) 2 a2 q The equations of the transverse polarized field in terms of the coupled modes are V E + = j - (a - a )sin z C V E - = -j (a - a ) sin D z. (4.50) Da2 4~ 2 q c The propagation constants for coupling between cyclotron and synchronous waves are Dq c' = + f 1 + + 4 (N/2)2 = + (4.51a) 2 y is real for all possible N, and 3 - +- 11 +41 + 4 (N/p2)2 = +, (4.51b) 4F'\ c 2 4 where TjV N C 2u w a o c A and A can be actively coupled, and A and A can also be 1 3 2 4 actively coupled.

-644.3.3c Electrostatic Slot Pump Field Structure. Figure 4.7c shows the geometrical configuration of an electrostatic slot pump field structure. The equation of the potential in the slot region is coth n n V = V + KV n sin z (4.52) s o n p coth na n n=l where K is an amplitude coefficient and V is the potential in the n o absence of the pump field. Consider only the fundamental component of the potential coth 8 x v V + V sin z. (4.53) s o P coth 3 a q q The equation of the transverse polarized field is E+ lV j2 X z -jp z i 2 q (e -e ). (4.54) E = J2 coth q a The propagation constants for coupling between A and A, A 1 3 1 and A are 2 1 +#] Jl+/l~+4 (w/)21, (4.55a) y 2 7 /j 1+ 4 + 4 (/), (4.55b) 4 where W - V pq 40 u cosh 3 a c o q In this case gain is possible.

4.3.4 Gain Computation of the D-c Pumped Amplifiers. To compute the gain, one needs only to consider the exponential amplifying and decaying component of the wave. One may write y z -y z A (z) = B e1 +5 e 1 1 11 12 y z -Y z A (z) = 5 e 1 + e 1 (4.56) 2 21 22 where B.. are the amplitudes to be evaluated from the boundary conditions 1j and c_ 1 +/1 +4 (M/2)2 2' c for 4(M/f2)2 << 1 c c Now A (O) = +, A (O) = P + 1 11 12 2 21 22 and from the original differential equations, B - j= Y j, 21 11 22 12 Solve.ij in terms of A (0) and A (0) and substituting into Eq. 4.56 A (z) A (0) cosh M z + jA (O) sinh z 1 1 Vc 2 cc A (z) = - A (0) sinh Mz + A (0) cosh z (4.57) 2 1 c 2 c The gain for the fast cyclotron wave is Gf = = cosh z = cosh o a2 (458)

-66For 2n sections, at the output 2nt 2nt nt ~Vpn Gf = cosh -, (4.59) xc a or the gain in db is nV pnt G fdb 20 log cosh (4.60) fdb 2 2 2 a2 c The gain for other types of quadrupole structures and coupled modes are computed and tabulated in Table 4.1. For the purpose of comparison, two sets of plots are presented in Figs. 4.8 - 4.10. In Fig. 4.8, the fast cyclotron to slow cyclotron wave coupled modes for different pump structures are compared. Here, the gains in db are plotted against the pump voltage for a constant set of parameters VO, PC a and n. It is obvious that the twisted quadrupole structure offers the highest gain for the same pump voltage. In Fig. 4.9, the synchronousto-synchronous wave coupled modes are compared in a similar way. Here the staggered quadrupole shows advantages compared to the twisted quadrupole. In Fig. 4.10 cyclotron-to-synchronous coupled modes are compared. Here, an interesting case can be pointed out. For example, in the case of using staggered quadrupole structure, (see Eq. 4.43), the coupling becomes active only when condition 44/p2 > 1 is satisfied. Under this condition, the gain is found to be G(db) = 20 logi cosh n Vpa2o2c )- 1. (4.61) The comparisons with other structures for similar sets of constants are obvious. -

-67w ~ ~ ~ ~~ 0 H 0 WD~~~~~~~~~ F-4 O0~ X~~0 w 0 r-w i —:o cr Fl- w o'O~0 0 0D 0 (~~~~~~~~~~~~~~~~ 0 E4 H4 If I- I a. 0 00 N E~~~~ci 0 EH II Cl) ~I 0 o 0 0 0-4 CD ~~0 -I 00 o i Hl 0 0 0 ~0 o aD 0 0 0 0 0 tor H qP'NIVD

-68o H 0 0 o~~~~~~~~~0 0o N I ~~~~~~~~~~~~~~~~u ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n~~~~~~~~~~~~bw I OM W-I~~~~~~~~~~~~I 0 W 0. 0 CfC) 0 w w> LO~~~~~~~~~~~~~~~0 F- OZ 11~w U? I w W u~~> a. ~ a o~I 00a II LzJJ {0 0 2 NJ 0 0L~ a.~a t- 0 w~~~~~~~~~~ v3 (D w Jw w > 0 I f E-o r-t 0 a. O I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I ~0~~~ H LA ~ ~ ~ L 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a. 0 / 2 0 00 0 0 co 0 - 0 o 0 0 pr; 0 ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~ 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ I —4 toC\ CI\

-690 0 w~~~~~~~~~~~~~ rr> w( cD a. w o ()dQ (-DJ 0O~~0 0 0 ~ H ~~~~~~u ~~~~~~~~~~I') II w Dw~o H I.uF-~~~~t. CC O a ~~~~~~~~~~~~~~o ~~ ~a P~o0 <f) 4N U I- w o~~~~~~~~~~~~~~~~~~~~~~~. C/) II) o -a. _ co.o W O', Ow~O 00 > j 0 L;E OE -i z 0 0 0 r1r)~~~~~~~~~~~~~~r>> E-i 0 2 ~o _o a. 0 ~ HI r o 0 H E-i CL~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C O 0~~~~~~~~~~~~~ O HI 0 0 000 CMW rO C'J~~~~~~ rz 4 qp'NI~~~~~~~~is9

-70Table 4.1 Gain Equations for Various Types of Beam Interaction and Different Types of Pump Fields Type of Propagation Constant Type of Gain Remark Pump Field for the Periodic Mode Copying (db) Structure B, 2a, Cyclotron- 20 lo l) Cyclotron cosh rV n3 rlVpn W2 a2 Staggered c Quadrupole| = 0 Synchronous- 20 log Eigh gain q Synchronous cosh for synchronous21rV L synchronous amplifier a2cw u c o O = O Cyclotron- 20 log(2)High gain q c Cyclotron cosh for cyclotron2rV nt cyclotron amplifier 2 2 Twisted c Quadrupole| B = 0 Synchronous- 20 log Synchronous cosh P a2w u c o q= P Cyclotron- 20 log(') High gain cRing q c |Synchronous cosh for cyclotronStrucinture ~2nrV n-r synchronous Structure p amplifier. amplifier. 02a2 Single field structure 1. Gordon, E. I., "A Transverse Field Traveling Wave Tube"', International Congress on Microwave Tubes, pp. 389-390; 1960. 2. Mao, S. and Siegman, A. E., "Cyclotron Wave Amplification Using Simultaneous R.F.-Coupling and D.C.-Pumping", International Congress on Microwave Tubes, p. 268; 1962. 5. Bass, J. C., "Microwave Amplification in Electrostatic Ring Structures", Proc. I.R.E., vol. 49, p. 1424; 1961.

-71Table 4.1 ( contd) Type of Propagation Constant Type of Gain Remark Pump Field for the Periodic Mode Copying (db) Structure (4) | = PC Cyclotron- 20 log Simple field Synchronous cosh Structure ~~S ~~~~~lot ~4V cosh3 a Slot 4o0 c Structure Structure = 2 Cyclotron- 20 log Cyclotron cosh rnVp 2V0cosh(2Pca) It is noticed that the plot of Eq. 4.61 brings out two important points. First, for a pump voltage below 50 volts (for the set of other parameters chosen), no gain is possible. Beyond this point, the gain rises very rapidly with the increase of the pump voltage. If this were true, then it would be an easy matter to construct a tube with high gain. Two points need investigating. First, is the condition 4M/p2 > 1 physically c realizable? Second, since the coupling in this case is between A and A and both of these modes carry positive kinetic power, and as both 4 modes are growing, and power must be delivered to them from the d-c supply, what is the mechanism for this power transfer? The next period shall be devoted to investigate these questions. Figure 4.11 shows the gain vs. 5 q plot for synchronous-tosynchronous wave coupled mode for a set of constant operating conditions. All curves show a decrease in gain as ca is increased. Gain is large for small 5ca. For a constant B, small B a means that a must be small so that the interaction field is strong on the beam. 4. Bass, J. C., "A D.C. Pumped Amplifier with a Two-Dimensional Field Structure", Proc. I.R.E., vol. 49, p. 1999; 1961.

-7210 9 8 5-_ 4 STAGGERED QUADRUPOLE TWISTED QUADRUPOLE 2 0 1.2 1.4 1.6 1.8,c a FIG. 4.11 GAIN VS. c a FOR SYNCHRONOUS-TO-SYNCHRONOUS TYPE OF INTERACTION IN DIFFERENT PUMP FIELDS FOR /V o = 4, L = 40 cm AND UO = 4.2 x 108 m/sec.

-734.4 Future Work. The questions raised in the discussion of the possibility of the coupling between a fast cyclotron wave and the positive synchronous wave are important enough to warrant immediate attention. We shall begin the investigation by correlating the results from other investigators to see whether such criterion has been met. An analysis of the total power relations in the system will be made to visualize the possible energy transfer mechanism. Other programs will include the continuing search for new pumping field structures, methods of exciting the various modes and eventually to study the noise configurations in the various modes. 5. General Conclusions (C. Yeh) An experimental low-frequency model of a frequency multiplier has been designed. It is in the process of assembling and final alignment. The device multiplies an input frequency of approximately 700 mc to an output frequency of 2800 mc. A feedback scheme is used to enhance the efficiency. The analysis of amplitude and phase-modulated traveling-wave amplifiers has been modified to take into account the multivalued nature of the velocity of the electrons when the beam is bunched. Boltzmann's transport equation is used to describe the kinematics of the electron beam. Difficulty has been encountered in brazing a tungsten helix designed for operation at 30 Gc into a smooth-bore BeO tube. Experimental results indicate that a material which has a higher resiliency than tungsten must be used for the helix. Coupled-mode theory has been used to analyze the transverse wave d-c pumped quadrupole amplifier. Equations have been derived for the

coupled modes of the different types of quadrupole fields. Gain in various cases are computed and tabulated for easy comparison. An unusual coupling scheme has been uncovered which might result in high gain and a careful evaluation of this mechanism is in progress.

DISTRIBUTION LIST No. Copies Agency 3 Chief, Bureau of Ships, Department of the Navy, Washington 25, D. C., Attn: Code 68IA1D 1 Chief, Bureau of Ships, Department of the Navy, Washington 25, D. C., Attn: Code 681B2 1 Chief, Bureau of Ships, Department of the Navy, Washington 25, D. C., Attn: Code 687A 3 Chief, Bureau of Ships, Department of the Navy, Washington 25, D. C., Attn: Code 210L 1 Chief, Bureau of Naval Weapons, Department of the Navy, Washington 25, D. C., Attn: Code RAAV-333 1 Chief, Bureau of Naval Weapons, Department of the Navy, Washington 25, D. C., Attn: Code RAA\V-61 1 Chief, Bureau of Naval Weapons, Department of the Navy, Washington 25, D. C., Attn: Code RMGA-1l 1 Chief, Bureau of Naval Weapons, Department of the Navy, Washington 25, D. C., Attn: Code RMGA-81 1 Director, U. S. Naval Research Laboratory, Washington 25, D. C., Attn: Code 524 2 Director, U. S. Naval Research Laboratory, Washington 25, D. C., Attn: Code 5437 2 Commanding Officer and Director, U. S. Navy Electronics Laboratory, San Diego 52, California, Attn: Code 3260 2 Commander, Aeronautical Systems Division, U. S. Air Force, Wright Patterson Air Force Base, Ohio, Attn. Code ASRPSV-1 2 Commanding Officer, U. S. Army Electronics Research and Development Laboratory, Electron Devices Division, Fort Monmouth, New Jersey 3 Advisory Group on Electron Devices, 346 Broadway, 8th Floor, New York 13, New York 1 Commanding General, Rome Air Development Center, Griffiss Air Force Base, Rome, New York, Attn: RCUIL-2 20 Headquarters, Defense Documentation Center, For Scientific and Technical Information, U. S. Air Force, Cameron Station, Alexandria, Virginia

No. Copies Agency 1 Microwave Electronics Corporation, 3165 Porter Drive, Stanford Industrial Park, Palo Alto, California 1 Mr. A. G. Peifer, Bendix Corporation, Research Laboratories, Northwestern Highway and 10-1/2 Mile Road, Southfield, Michigan 1 Bendix Corporation, Systems Division, 3300 Plymouth Road, Ann Arbor, Michigan, Attn: Technical Library 1 Litton Industries, 960 Industrial Road, San Carlos, California, Attn: Technical Library 1 Dr. R. P. Wadhwa, Electron Tube Division, Litton Industries, 960 Industrial Way, San Carlos, California 1 The University of Michigan, Willow Run Laboratories, Ypsilanti, Michigan, Attn: Dr. J. T. Wilson 1 Microwave Associates, Burlington, Massachusetts, Attn: Technical Library 1 Microwave Electronic Tube Company, Inc., Salem, Massachusetts, Attn: Technical Library 1 Radio Corporation of America, Power Tube Division, Harrison, New Jersey 1 Raytheon Company, Burlington, Massachusetts, Attn: Technical Library 1 S-F-D Laboratories, 800 Rahway Avenue, Union, New Jersey, Attn: Technical Library 1 Tucor, Inc., 18 Marshall Street, South Norwalk, Connecticut, Attn: Technical Library 1 Dr. Walter M. Nunn, Jr., Electrical Engineering Department, Tulane University, New Orleans, Louisiana 1 Westinghouse Electric Corporation, P. O. Box 284, Elmira, New York, Attn: Technical Library 1 Bendix Corporation, Red Bank Division, Eatontown, New Jersey, Attn Dr. James Palmer 1 Mr. A. Weglein, Hughes Aircraft Company, Microwave Tube Division, 11105 South LaCienaga Blvd., Los Angeles 9, California

.0c~ ~ ~~~ 0 i00 0) ) 4) 4a) HH0.-, ~. AHO 0- 1- 0 14~~~~~~~~~~~~ B 1 4 =1 Me PIb9~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r,P,1 P,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. 00 o00~~~ OH. - ~0 000 0) 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/ fr.0.44 H1 OH H H 4Ah.00 H 00 0)~~~ 4) HO 000 0 0 0o OH 0 oH 4 0 00 A4-1 4) 4; H HO 0 0 0 ~~~~~~~~~~~~~I~~~~~~~~~~~~ ooOH 0 0. HHo H o 0) H H0 CD OI, "141 - Id PI P00oo -lo A HO b HO. 0 04- 0. U0 Al ~ 1 00 00 000 HH OH 0) 4.H HO~~~~~~~~~~~~~~~~~~~~~~QQ HO. 04-.Q 0.00 HO) 0). 0 00) 0 ) ). 0.0.00 co OH H 0 4 0)H 0 H O0 000HO 00 F-I H10 0. HO 0, 00 H OH0 04-40 H 0 )0 ~~ 000 HOO) 0 00)0 O)04-4 0 0 00~~~~~~~~~~~~~~1 1 4-O)) H g O4W) 00 0 0 o AoH 000 0-i 00 00.0) OHO 000 000 0)0 4-HO 0 0) 000 00.0 0 0 00 0)0~~~~~~~~~~~~~~( 000 0 4000 0)oOOH H 00.2 000 0.000 0 0 00'-'- 0 00H H 0 HO 0 0004)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~11. 9 o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~w P 6~~~~~~~~~~~~~~~~~~ rl~ ~~~~~~~~~~~~~~~~~~~~~~1 4)004 0.0 000 H 4-0)0 4- 000)0 0).00H H HO 00 0~~~oo0 4-00 OH~~~~~~~~4 0) Ho 0 00 0P oo o 0 0 - 0 0 00 0k 4- 00 0) H 4-0)0 0- 0 O 04 —. 00), H 0 4 40). 00 00 00 00 00H 0 00000 00 00~~~ ~~~4- 000;I0O H000 OH 0 00 000)4- 0)00 0 4-00. 00~~~~~~~~~~~~c 0000 000 00 00- 4- O O al k d 9,~~~~~~~~~~~~~m 4 C H A 4 cdrl 14 d4 0 4 0 H 0 0 H 00 004 0 400. 14 -P0 040 00 0 C) 0) 0- 0000 000 HO 00~~ ~~~0 0)4 04-1 0 Ho'-.0 00 00 00 0 0 00004 00) ~~~~00 4-0 HO 01 I 00; 0 0 H 0 00 -) 4- 0d 0 OH 0)4-0 000 0 040 0. 0 0 000 0 H 0 0 0 0 4-0000 - H 0 o 41 ~ ~.,0 4-0'- 44 " -40 0 0 4) ~ 4 0 0 O 4-0 HO 04 0 0 4-00 04- 00 00 0 0 o.H 00 1-1 11 -4 0 0 4 0 -H 00 0-) 0 4Id00 0 ~c tI D0 ) a 0 H 0 4d o 000 ~~ p 0) H -.40 0)00 00 400.o ) 0H 000 OH -4 000 00 -zt -4 4-1Z 4401 4- C)-4 44-4 4-4 CC4. 4 0 4 0 a 9 - (d o d od 0 " ) b 0\ 0 0 0000 0.0 0.0Q,P4 1-0 I, ZO 1 I, o 0 4 00 40 00 00 0 00 4 -4 I 00)0)0 00 000)0 0 b 5' 44,24 0 4)4~~~~~~~~~A0 0-04 4-000) 00) 0'4-000) 00 0.4 Idb, o Aco ) P 4 C),2 44 0).54,2 104 C.)W' 00000) 04- ~~~~~~~~~~~4O 0 H 00000~~~~~~4-Ad 01 P, N 9 " 0 Id~rl, ~ a, a)442 0 0004 -0) l, N 00)000 Id 0 00)4)0 to C) 4 O"~~0,2- 00 OH H ~~~~~~~~~~~ 4 HH ~~~~~~'-~~~~~ 0 4o4~~~~~~~~~ 444ooS1~1)'AQ 0 00)00 04- 4-44-u44. 4- 0 0000 8 a k( d I~~~~~~~~~~~~~~~~~~k- I i t r 8 r o a ~~~~~~~~~~~~~~a~~~t,q8 HOo).0.0. ooo 04-000000004- 000000004-000000004-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~44-4 0 0 4 04-i 4-4 H 00 0~~~~~~~~~~~~~q H4 00 0 ~~~~~~~~0000~ 0)0I~~ 00)o~~~~ 40m~ 000k~ 0-0 0Cc4' P K40 0 4)0)d 000 400 oo 00 ~ oI O 00 -.o o 0)0 4 04-U 0) 0) 00 00- A 4- 0 oO) 0O..0 0 00) 0~~~~~~~~~~~A l )0 a) 4100 H Id0 ~ 04 4O 0 0a 00 0 4- a) 04 4 440 40 00-0 O 4 00 h 4i,2 a) d o 4-0. o o~ c kQd~-, o Q) 0) 00 4 0) 00 0 0 ) -P 0) 0 0 H H a))o o 0 0.4- 0 0) 4 0 0 00~ ~ ~~~~~~~~~~~~0 45044- 41 - 0.0 co C- 0.0 O -4 0 0 00 00 0HO 00. 0,4 40 0. 000 0) 9 004'-; 444 4t 0' 040 E-10 44 0 0- 0 oc 0 0) 0) 00 00 0 0 0 0l 0 4 54 00 -,4 ~ ~ ~,. z d d~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~d El. O' 00 - P o.'oo 400 0 000 0 P,2'- 08 0010 400 40 )0 4 4.oo m~ 0'd 4-0 4 00 0)00 to la C,'Plp,a, (l 14c -0' 41 4 0000-0 OOA4 4 45 H50) o, Xa, rlc, k Flr~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ a~m dr'-HO 00)0~ ~~~~ ~~~ 000 4-4 044-0 0 4o.4C)4 to OFH -1 d -R a c~ Ab0 40 HO. 0 40 -00 0)0.00 b.4A.4o 0co00 4- 0-0 0 00 84- 0)0 0 PQ 00 0 0 4.4 0O 0 401 - 05 ~~~~~PI rrl hO k r: (d~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0Ia 0w O 04a~~~,4,, D P 4- ~ ~ ~ ~ ~ ~ ~ ~ 4-4 0 (d al drl a~~~~~1 d ) - 00 0 0~0) 0 -4 0)..~g 50) 4a4 440 a-o 00 40 04 a a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~41g ja a 0 ct- 0 W. 04-0 HA-0 L00 0)4 04- 0, 1 4 40 001 04Z0 5. 04 O 0 04 0 0 O 0'\ 4.444 4 t- 4- 0 0 co 00.00)6 008c544 i 000 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~0~~~~ 4~400 4~4 040 H 0).4 H ~04 ~ oH,2 4-000 oo d,2.,A 40 P0 00 041l-1 ~11 40~~4 4-0 00) 040 - U20 0 0 0 - 0 4 0 )4 IN,~o 0.4.44-4 41 ) 0000 00 0) 0 40,2 4-004 00 4000 4..% 2-~~~~~~~ 440 0200 4-~~~0WWO 004 04. 04- ) 4o74 4) Al 544> 0U) - 4-0,2 N'\ 4- H00~~04.- 00 )5OOj 04) -40-0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~- ) 0) 0 4 )4 4-E0) 00 4-0)0~~~~cd 0 0 0 ) 0-040 0 4 4 44441 4.44 444,2.41 -~+~.4 400).0~ 4~ 004 05~ 4AH ri, k ux 0 H 4-~~~~~r da -P o 14 91, r r( 1, OU dd (uam(d k ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~aa~~c 2 dC 00 0 8I ) I to b,' bo 0 Opr 00 ~~~~N n I ~~~~~~~~~~~~~~~nH~~~~~~~~HHi;S~~~~~~~q D0) 41 I 1'O00 4- ( d WK P. z4 O 8 4- P:, EA 4-$X a, k. k "d Q) (d~~~~~~~~~~~~~ C ) - CU;-~~ +C co a) (H X ~~~~~~~~~~~~~~~~~~ k~4( Po P-e 0 ~~~~~~~~~~~~.t?'~~~O 0 9 k

P. j 0. H'0. * ~.0 o 0o i. ~ o'ooO'- P, 0) Oa) 4- O' Ho00 o ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o P ) 4 L o 5c oa)0 H~2V'O.0 O0HHS -,. HW0. ~~0.5~0O0O ~ ~ ~.~O 0 i~~5~ 00010 HHOO~~~~~~~~~~~~0 -0100 0001 ~00 O) A "0 00 0'0410 0000~'''0.0 000,0 H'000001~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-4) 140P,-j*W01000 Id 0 00WW 000 00 cdNO o o+o' 0 60010001O00 0000 00100 0'w 0~ OHO HO 00 0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.r - 0. 0 0 01 HcOHoH 0.0 o H 0 0O 00 H H o 00 O ~'O oo 0 I-)0 4 0.5 ) 000 0 01104-1 0 H P, 1HO 0 0-P01 0 O0. 0 H.3 01C) 00- - L - L (u 1-i ~ ~ ~~00 0 Hr 1a 00 00'0; I 0 0 OOH H ) -0H i9 ~~ 000000 00 00001 I~~~~~~~-1 Hd:: 0L H0)4 O H)d000 0 la) SU $1 Cd'd 0 0 -P,~O Go HO\ 4-1 4 0.a0 100, 0) 000001 0 0a 4HOd F OH 000 O 4-) 1041 0 ('I O 4 0004 0~~~~..~' 505 HOW 0,0101 0~~~~~~~~~~~~~~~~~~~~~~~~~4 H) 00 00000011 00 00110i 00 co 00 H0004, 1- 1, 00 H O 000 0 0 04110 00 4 O 4 ~~~ OO 0I' H H 4004 O v 2 00w H H HO'0. 510 00 5 0~ -015Id r. HO0 00:S 0 o i r' 00 H' 0 0CE- A 000.o 0H 00 O 1 0~ I 000 0H4-0, H41 OHO~~~~~~~~~~~~~~~~~~~~' HA00 0000 0000 co 0,1 0d 0El 0 0. 010 00400 0~ 0) () 00C 00 - HO) 000P U H 00 000 H HOO'00~~~~~~~~~~~~I0 0l a) 00 0 0 25.C0.)O Hi 4'0 HH 20 4OH.~~~)000 0,,55 000 H 0100 01 0000~~~~~~~~~~~~~~~~~~~~~~~E 000-1- H HO0 I'000 0) C00 0 0 00 00 041 0W 0 01 00 G o 0 OO 4 H 000 0 0 H 0 00 01,O'o1 1 t0~ 01 0 0 001 0 0.4 10 o0 0000 OH 00 0100 0 O~~~~~~~~~~~~~ ~~ 00 H 0 000'0 0-9 51 510001~~~~~~~~Cd -H 10000 00 H01~~~~(L'00~ 0)' 0 35 0 0 O 00 0100 01 00 11) P ~o oJ. 0 Id 0100 01 041 H. 0 0 0'5 11500 00 H HO H O 0~' 000 00)01Eo''0 0 01' 00 a)0~' 0010 41 00 I'd 0 00 to 00-0 015.o 04Ii0 0.-i o 0 4100 0) 000~' 01 01000 4101 ~~~~~~~g 00 o'0 H 00 40- 0 0 10.0$0 0.010 0 40 W 04 100~ Id 00'00 00 01'0 0.5~~~~~~~~~~~~~~~~~~~~~I 03 H 0 0 0 0 0 1 0 1 0 5 d -0) H 00 00 d 0 4 31' 00 0,, - 0 4100 -~ ~ 0 D0o10'A0 0.10 H 1i L 000 d 00 Ea 4O S00H0 01 0~' 001 0, 0 ~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 OHHOO~' a)0 0 511 u 0) 01 01 01; o 0 00 001 00 ) 01 ooo0 0'0 00 00.0 000110 ~~~~~~~~~ 010100-1000Q 41 0101 4- 4 d41( 0H10 01. 40 0 0O0' 0) +Q) " 0 00l0 000 011 0010001 400 P 55. 4' 0 00 0 40 00 0 0o 0 0 00.0 00 0010 0 HO 0 0000 01 00100 o 00 0. 0 0000 00~'o 401 0 H 00 0 0 00 H 0 0 0000 0~' 0010 000 0 4I 00001 0010 W o o -H o~ Id o 000 0.0 Id 0to 0- 0 - 0 00P,00 A1 000 oHC 0 0- 0100 H ) 000 01 0 5151~~~~~~-~~ 1051 51011-0 00.5 51 51~~~~~~~~~~~~~~~000 WI OHa)U 00 010 01 Id 00001d co.3 4- 11 " a) A51 105 0- 040 0 0 000 ~~~~ ~~~~~~~ Go ~~~~~~~~~+1 o - id; - 4- Ida01 - O bo:J 1-4 ci 5- 41 0 ~~i >4.01 00 00 Q) 0) ~ ~ ~ ~ ~ 0 a) 0 10010 00 000'0oO0 5, o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~E o50041'00 10 HP,k1i ~ ~ ~ ~ ~ ~ ~ ~ 505.0120 0001000 ~~~~~~~~~~~~~~~~~~~~~~~oo~'oooo 0000 0001000000~' 00000000~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4H 1 H.55'1 0 0 5 1 ) DO H 00 o 00 00~~~~~~~~~~~~~~o 00 555 - - 00d 0a0) 0 0 0 0 00 0 0H.5w1 1 00.0A. d~4''H... 0 0 0 01'5,0 ~* o ~' 01 0 0 0~~100 1 0 511 1. 00P-.00 5 00 51s00 I~~~~~~~~~~~~~~~~~~I O~ 000 0 50154o HO 00 10 000 00~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t 0000 0 Ho 0~40 000 00o 00041X 0001 P 00 10 0) 0. 00 01 0 0 5A~' 0 00 000 0 0'1-10 0pioo 51001 0 0c 00 000 0- 01 00 000 5000 d 0 t)- 0 00 H 0 I D 0 100)C)C\ to10;~ 0 14 ci 40 00 0 0 a) 0- 0 0 000 10 0 0~~~~~~~~~~~~~~~~~~~~~~~1 00I01 0 0 41 041' H. 1 -( 01 0D 041 ) 0a4.5, 0 0 1 ~~~~~~~~~~'~' 0 1 0 0'1 0' 040 00' 0 0 01 ~' 01 00' 0~-0S o~~~~o 0 ~~~~~~~~oo 5,5 51 5.0 0~~~~~~~~~~~~~~~~~~~~~~ 0 001 0 0.0~~~~~~~~~0 0 (D'oq H o 40 00 0 00 4 0- ~ 0 0 00.5~ Quo 0 001 o3o51 01 0100000000001 00100~~~~~~~~~~~~4 H c 01.0 0 0 0 0 W 0 0 1 0 0&: 0 0 0 0 0 0 0 00 55 bD 0 A.51'd 5. a)011 D OW 001 00' 40.0> " 01'5 0 451H0 0 00'0 5;-.5 4-l Id15 01510 0100 0 01m- 0';I q1 oi 010 0 0 0 5 0 00T 00 00q 010 0 000 5A 5 5551 0' o 4- d l- 4 H 4 0\ o:\ 10 00 0 U 50

UNIVERSITY OF MICHIGAN 1III 5 022 7IIllll II lli i1I 3 9015 02526 1671