HE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN QUARTERLY PROGRESS REPORT NO. 4 FOR BASIC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONICS Thi{is report covers the period February 1, 1964 to May 1, 1964 Electron Physics Laboratory Department of Electrical Engineering By: H. K. Detweiler Approved- by?: - M. E. El-Shandwily C. Yeh J. E Rowe Pro,ect Engineer J. E. Rowe C. Yeh. Approved by:: /.... J,. Rowe, PDirector 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 May, 1964

2' 11', h-f' i ji t 5

ABSTRACT Generation and Amplification of Coherent Electromagnetic Energy in the Millimeter and Submillimeter Wavelength Region The construction of the experimental low-frequency model of the frequency multiplier with feedback scheme to enhance the transfer efficiency has been completed. D-c beam testing shows that the tube is aligned satisfactorily. The tube is now awaiting final r-f testing. Difficulties of brazing a 30 Gc helix into a BeO tube have not been resolved. Techniques involving the use of Fansteel 60 metal are being developed. Analysis of Amplitude and Phase-Modulated Traveling-Wave Amplifiers A nonlinear theory of the amplitude and phase-modulated travelingwave amplifier with a multiple frequency input is developed. Computer results for some typical cases are presented and interpreted. Study of a D-c Pumped Quadrupole Amplifier The mechanism of an anomalous gain predicated for the cyclotronto-synchronous wave interaction is explained in terms of kinetic power relations, Experimental results from other sources give support to this theory. -iii

TABLE OF CONTENTS Page ABSTRACT iii LIST OF ILLUSTRATIONS v PERSONNEL vi 1, GENERAL INTRODUCTION 1 2. GENERATION AND AMPLIFICATION OF COHERENT ELECTROMAGNETIC ENERGY IN THE MILLIMETER AND SUBMILLIMETER WAVELENGTH REGION 2 2.1 Study of Frequency Multiplication in an Angular Propagating Circuit 2 2i1.1 Assembly of the Experimental Tube 2 2.1.2 D-c Beam Characteristics 2 2.1.53 Future Work 3 2.2 Investigation of High-Thermal-Conductivity Materials for Microwave Devices Above X-Band 3 2.2.1 Introduction 3 2.2.2 Experimental Effort 4 2.2.3 Future Work 6 3, ANALYSIS OF AMPLITUDE AND PHASE-MODULATED TRAVELING-WAVE AMPLIFIER 6 3.1 Introduction 6 3.2 Derivation of the Equations 6 3.3 Computer Results 18 3.4 Future Work 23 4. STUDY OF ATi D-C PUMPED QUATDRUPOLE AMPLIFIER 23 4.1 Introduction 23 4.2 Kinetic Power Relations in Cyclotron Synchronous Mode Interaction 24 4.3 Experimental Evidence of the High Gain Transverse-Wave 28 5. GENERAL CONCLUSIONS 28 -iv

LIST OF ILLUSTRATIONS Figure Page 3,1 Gain vs. Normalized Distance 19 3.2 Gain vs. Normalized Distance 20 3.3 Gain vs. Normalized Distance 21 3.4 Gain vs. Normalized Distance 22 4.1 c-p Plot of the Cyclotron-Synchronous Wave Interaction in a Twisted Quadrupole Pump Structure, with M > p2/4. 26 __ c~~~~~~-

INTERIM SCIENTIFIC REPORT NO. 4 FOR BASIC RESEARCH IN MICROWAVE DEVICES AND QUANTUM ELECTRONICS 1. General Introduction (C.'Yeh) The broad 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 design described in Quarterly Progress Report No. 3 will be constructed. Extreme care will be required in the final assembly of the tube particularly with regard to the alignment of its different parts. D-c beam testing will be conducted prior to the final r-f testing. B. Investigation of high-thermal-conductivity materials for microwave devices above X-band. Work will be continued to develop a helix loading technique. Other suitable helix materials, e.g., Fansteel wire which has a higher resiliency than tungsten, will be investigated. C. Analysis of amplitude and phase-modulated traveling-wave amplifiers. The equations presented in the previous progress reports, Nos. 1, 2 and 3, are being programmed for digital computation. Results will be presented in the forms of curves followed by discussions. D. Study of a d-c pumped quadrupole amplifier. An anomalous gain mechanism for the coupling between a fast cyclotron and a synchronous

-2wave was discussed in a previous report (No. 3). Attempts will be made to explain this type of coupling mechanism by means of the kinetic power theorem. Experimental evidence for this type of operation will be sought. 2. Generation and Amplification of Coherent Electromagnetic Energy in the Millimeter and Submillimeter Wavelength Region 2.1 Study of Frequency Multiplication in an Angular Propagating Circuit (C. Yeh, B. Ho) 2.1.1 Assembly of the Experimental Tube. The assembly of the experimental tube has required more time than anticipated due to several modifications incorporated in the multipole magnetron cavity. First, the pole pieces were removed and replaced by coupling flanges onto which straight glass tubes were attached. Then the r-f coupling loop is rearranged so that the cavity can be fitted into a focusing solenoid. The final assembly was quite complicated due to the need for extremely accurate alignment. One addition to the original design is an ion pump attached to one end of the tube. This addition is necessary in order to maintain as high a vacuum as possible during the life of the tube. 2.1.2 D-c Beam Characteristics. One of the first things to check out before initiating r-f testing is to measure the d-c beam characteristic along the tube. The tube assembly is carefully centered in a solenoid which is capable of delivering up to 1000 gauss of magnetic field. With the input coupler, multipole cavity, feedback coupler and the collector all connected to a common low-voltage power supply, but metered individually, the gun voltages are adjusted for a predetermined magnetic field strength determined by the required cyclotron frequency, until all the electrode currents are zero except that in the collector

-5circuit. A collector current in excess of 200,la can be obtained. It is then certain that the alignment of the tube is satisfactory and the beam interception is negligible. The d-c operating conditions are D-c magnetic field (30 volts, 1 ampere) 250 gauss First grid voltage 4 volts First grid current 600 pa Anode voltage 25 volts Anode current 50 Ba Input coupler voltage 25 volts Input coupler current 0 Multipole cavity voltage 25 volts Multipole cavity current 0 Feedback coupler voltage 25 volts Feedback coupler current 0 Collector voltage 25 volts Collector current 150,a 2.1.3 Future Work. An r-f signal source capable of delivering up to 10 watts at the signal frequency of 680 mc is being constructed. It is expected that r-f testing can be conducted during the next period. 2.2 Investigation of High-Thermal-Conductivity Materials for Microwave Devices Above X-Band (H. K. Detweiler) 2.2.1 Introduction, Efforts d.uring this period have centered on finding a way of overcoming the difficulty encountered in loading a helix into a smooth-bore BeO tube for brazing. Several means are presently being investigated. A detailed description is given below.

-42.2.2 Experimental Effort. At the end of the last period the point had been reached where it no longer seemed likely that a heat-treat cycle would be found that would give the tungsten helices the necessary amount of resiliency without excessively embrittling them. An inquiry has been made into the availability of BeO tubing possessing greater dimensional uniformity so that less clearance between the helix and tube is required for loading. Unfortunately, more time than is desirable is required to obtain tubing with the necessary tolerances. Consequently, this will be pursued further only in the event the other alternatives fail. An investigation was made into the possibility of finding a suitable helix material having a higher resiliency than tungsten which could be used with the BeO tubing presently on hand. A high-temperature spring material, Fansteel 60 Metal, appears to possess the desired properties. This material is an alloy of 90 percent tantalum and 10 percent tungsten prepared by electron-beam melting techniques. Since some of this wire was immediately available in a diameter of 10 mils, initial tests were performed on this size. A suitable helix was scaled for this wire from the 30 Gc size. The dimensions of this helix are Mean helix diameter = 0.065 inch, d = 0.010 inch, and w TPI = 32. A cycle for heat treating this helix was developed which resulted in it being possible to wind the helix on a 0.050 inch mandrel (5 mils undersize) and have it return to its original size (0.075 inch O.D.) * Fansteel Metallurgical Corp.

-5when released. The heat-treat cycle is to fire the helix in air for 15 minutes at 5380C in order to oxidize the surface and then fire for 30 minutes at 12000C in a vacuum to diffuse the oxide into the metal for hardening. After testing of the spring properties of this wire, tests were run to determine if it was posslble to plate it in the manner required for brazing. The outcome of these tests was positive. After conclusion of these tests it was determined that, if the smaller diameter wire behaved in a similar manner, it would possess sufficient resiliency for the loading process. Attempts were made to obtain the above material in the required 5 mil diameter. However, none was available. Consequently, some 5 mil diameter Fansteel 61 Metal was obtained for testing. This wire consists of 92-1/2 percent tantalum and 7-1/2 percent tungsten prepared by the sintered-metal process. Unfortunately, its behavior after heat treating was similar to that encountered with tungsten, i.e., it is either too brittle to wind down without breaking, or it is too soft to spring back a sufficient amount. The 60 Metal wire is presently on order and will be tested upon its receipt. Because of the difficulties encountered in this method of preparing this structure, a somewhat different approach is also being investigated. This consists of brazing three BeO helix support rods to the helix and then pressure loading them into a smooth-bore copper tube. The loading technique is presently being developed; the BeO rods are on hand and the copper tubing is slated for delivery in the near future. It is anticipated that a heat-test model of this type will be prepared and tested during the next quarter. Some work recently completed comparing by means of r-f cold tests the electrical properties of helices brazed into BeO cylinders to those

-6of pressure loaded helices has shown that the performance of the helix is not appreciably degraded by brazing, Consequently, assuming the same thing to be true for the 30 Gc structure, only the heat transfer properties of this geometry will be investigated. 2.2.3 Future Work. Work will be continued on the techniqyue of loading the helix into a BeO cylinder when the Fansteel 60 Metal wire is received. The BeO rod-copper tube structure will be assembled upon receipt of the tubing. Both structures will be heat tested pending their successful fabrication. _. Analysis of Amplitude and Phase-Modulated Traveling-Wave Amplifier (M. E. El-Shandwily, J. E. Rowe) 3.1 Introduction. In this report a general analysis is given to describe the nonlinear operatlon of the traveling-wave amplifier with multifrequency input. The final equations are solved on a digital computer and some of the result;s are given. 3.2 Derivation of the Equations. The equations that describe the interaction between the circuit and the beam are as follows. The circuit equation is az2 v2 t2 2 E at2 O O O The force equation is lo Detweiler, H. K., "Applied Research in Microwave and Quantum Electronics", Final Report, Section II, Electron Physics Laboratory, The University of Michigan; March, 1964.

Z + % (3.2) The conservation of charge is wrltten using a Lagrangian formulation to account for the crossing of electrons, i.e., the beam is assumed to consist of finite particles and the conservation of charge is obtained by stating that the charge contained in a bunch at one position z0 and at time t 0must be conserved at some other position z and a later time t. Hence P(z0o to) dz0 = p(z, t) dz o (303) Due to the inherent nonlinearity of the system, it is expected to find. components of the circuit voltage, charge density, and current at the haronic frequencies and cross -modulatiSon frequencies of the input signals together with those having the fundamental frequencies. In view of the above the expression for the circuit voltage will be written as 1 II I Zn -j%n(Y lol ) o n An(Y n1 V(z, t) 1 (7Z) 2 C A n= - oo where y = C z /u, i 1 O w is the frequency of one of the input signals. BeIore carrying out the substitution of Eq 35,4 into the interaction equations, one must find a relation between On} no and z. This relation is as follows:

-8(c c en(Y-) (= nt n+? (y,5) 1 1 Using the definition of y and Eq. 3.4, the left-hand side of the.ircuit 1 equation becomes ~ C 2 * Z d2A ( d dO (2 0 Uno L on y2 dy 1 n 1 n n 2.n 2 n n 6 n n A t1 1 1 n n c n ensaal zi n f tn II 2n C d nd Z will be written as 2P = 2 n) e 7) v n n n series since inthe rf chargensity in the beam is expected to contain components at all possible combination of frequencies the charge density will be written as = j (Yi) e n (7) n It should be noted that the above expression for p is not a Fouirier series, since in general the n's are not harmonically related. It`s simply a convenient mathematical expression for the charg~- dens.ty1 The right-hand side of the circuit equation becomes L n 22n J 22 nC d g.) n1 Un

Equating 3.6 and 3.8, one obtains 2n d2 Z rd2A dA w dO d0 w dO (Cdy2 dy C o dY n 2 C d n y dY2 1 i 2w2d Z 1J + 2 n n n n n n n n n Uo n 2 2 n n (n n 1 n |IoI ( C1( D1 ) n n The above equation can be separated into n equations by equating terms of like phase on each side of the equation. This can be justified mathematically as follows: If the time average of Eq. 3.9 over a certain period of time is taken (after multiplying both sides by e n) and taking the limit as T * oo, then both sides go to zero except when n = m. From the above, it is seen that Eq. 3.9 is valid for each term in the summation. p (y ) is the complex amplitude of the component of charge density at frequency wn, and can be written as Pn Pnr + jpni Equating the real parts on both sides of Eq. 5.9 (after dropping the summation) and letting Bn =.xn/C X gives

.10l n n B22 dyA (n 1 dy'y ) B n U U U 2 o o 2 =CB (3 lo) fl nv En v I7 n nn n XI 0 fl Q Equating the imaginary parts on both sides gives [d28 n d A (Y dO d. A,Y da o &a 2 0 -n n An(y) 2C d BK ( )-] A n. n 1~~, [ dy2 n n n' 2 dy' n d 2 U U u u =C o o 2 - o (5,1o) n Bn v jI Pni n v I 2Cnn n nr n 0 n 0 Equations 3.10 and 3.11 give 2n circuit equations, To find the amplitude of the charge density pn, multiply Eq. 3.7 by e on both sides and take the time average over a certain period T, T T m 1 Using Eq, 3 7 to express 0 and 0 in terms of cot, 0 t and in the limit as.;-T -, o. the' right-hand side goes %Q zero unless n ~ m. This is T Tlhe ilnt~egr,ation can be tr"ansfonaed. to int~egrtationlk overl phase so that$~

( (t=T) 1 (t=T) -L Tim T sin e mid, (3o13) p - Lim T -, p scos m dm. (3014) T — > m ~( t=O) m The relatiQn between variables are found from Eq 31,5,~ + 0 (y) = B y -Bt n n 4_ rn Differentiating with respect to t gives 60 60 dy dy n ~n B dy dy dt n ct n i 1 The velocity is defined as dz/dt = ut- uo(1 + 2C u), so that dY0 dy cw (l+2Cu), 1 1 1 ( 8r n)(C + 2C u - 1) _2 n n 1 1 n 2u nB (1 + 2C u) ( 1 + 2Cu (y315)

-12 Now, it is required to express the charge density p in a convenient form. This is accomplished by using the continuity equation and the relation between the variables, Eq. 3.15. From Eq. 3.3 p(z,t) = P(z, t) dz and assuming the beam enters the interaction space unmodulated 101 zo p(z,t) = - u. Define a distance in terms of the time of entrance of a certain electron Zj utj also define = o w j, where oj is the phase of the wave propagating at frequency w when the jth electron arrives at the interaction space. Thus u 0 zoj = o lj and 0 0 01 _ 01 aZO t uO a1>; Since' is a function of the normalized distance y and the inirtial n 1 phase of the electron, then the total differential of On can be written as d~(y,, 0) = 1 +' 1. O1 1

but from Eq. 3.5 dn(Y' n 01 = Bn dyl - wdt -de (y ) and therefore (Dr) + y( n _01 B )o dy dt n dy (nd t 1 01 t im: 01 1 01L 1 1 1t 1 n1 1 + 10 1 p(z, t) ='y i = + (5.16) 0 1 1 n 01 y The 2n circuit eluations nitow take the following form: d2A(z,) de(Y' 2 ) = U 1 n1 dAn(Yi) (1 + Cr B - (1 2(1 C b ] dy2 n L \n dy / a nn dy2 2 C(os Bn. (y7, 1+ ) 11 01

An() [ d B (1.) +Y) 2 C b B - d(1) Cn n2 (1 + Cn bn) n n d2 n d B dy n dy a1~ 1 01 B1 + 2Cu(y, - d 1 1 01 n 1 01( where the definition uo/v 1 + C b has been used. The force equation dt2 [ + -]will be written in terms of normalized variables previously defined. The following relations are derived: d2y 1C)c d2U dt u0 dt d2 2CXw' (1+ 2u) u, dt 11 1 1y (1 + 2C uC )[ aZ -"v i 1 1 0 Using the expression for eV/az previously derived gives

-15(1 +2C u) - n e dy An Bn- dy 1 1 y4C un B P ( zj) B i O n 2C o u SC 1 It was shown by Rowe that the space-charge field can be written as p(z) R2 E j If the expansion of charge density is introduced, one gets: R2 -j(D E nn e n SC = Ab2 2 n sc bcb' Using Eqs. 3.12 and 3o16, Iof MihgnR2 Apri 1 Sc E~b12U CE ~, i n Tn -J on 1 _ 31101 (1 + 2 u) 6'1/D n Etb 2U a 1 1 T-+oo T (1 + 2C u)

Taking rn =- n/v and using the definitions, 2 0 cn IiI aP =) b'2 Z 2 P ceb 2u Zn 2u 0 0 the force equation becomes 2 c z(+ u)C dO au 1 n n n n (1 + 2C u)1 dy An dy j ] P n n 1 (n +- Lim d(D 3o19) 2_ ( X X 1 + C b Lim T 1 + 2C u d 01 i n In the separation of the circuit equation into n equations the time average has been taken over a certain period and the separation has been possible by taking the limit of the time average as the time interval goes to infinity. A limiting process is necessary, since in general the input signals have no common period. However in the special case, when the frequencies of the input signal are commensurate, there will be a common period and the time average need be taken only over that common period. Assuming that the frequencies of the input signals are given as w = L z, w =, LA - LNA /,(where L, L,... LN are 1 1 2 2 1 2 integers and Ao is the greatest common factor), then the common period is equal to 2r/Aw. The four working equations can be written as follows. Circuit equations:

-17dA (yl) - () L ( dey1(y) ( B2 (1 +C b)2 n n 12 n A n(yl) Bn dy. n...Cn n dya 0 1 1 01 2icL s in ((y, a j 1 + 2C u(y, o ) (.0 0 1 1 01 A1() d2e n ) - 2 2 1 dA2n(y1) do (y)) 2 (y 2C d aB(1 +,C b B - 2 dy2 n 12 n dy d.y 2~rcL 12 1~~~~~~O 2 /n1[ 01 C B (1 + Cn b 2~r S sn nnY,~ l n- Bn nb) gL 1+ 2C u(y, 1 do n n 0 ~~~1 1 01 0 2x dn(Yl ~d) 1(3 1 + 2C u(y, (. 0 1 1 01 Force equation: Z dA z do 6-U1 n n n (1+ 2C u) s) co0s o- a sin 0 (B An z C ay C n \a dy 1 1 n % 1 11n 2'2 ""2 B:+Cb2 -' s in -4 - dy 1-b I~ / sin (~ (Yz 2 w 1 + c b 2licL 1 + 2C u 01 c~~~~~~~~~_ o n n n 0 1 1 o 22)

Relation between variables: ( dO c 2.+ =- (3.23) dy dy 1 + 2C u 1 1 1 1 3.13 Computer Results. The equations presented in this report have been programmed and solved on an IBM 7090 d.igital computer. Some of the results are shown in Figs. 3.1 through 3.4. The input consists of two signals at frequencies f and f. It is assumed that seven 1 2 signals will propagate on the circuit. These signals have frequencies f f, f = 2f - f, f = 2f -f f 2f, = f 4 f, and 1 2 3 2 1 4 1 2 5 1 6 1 2 f = 2f. Only the first four are shown. The output powers in db of 7 the signals at f and f relative to their input power are plotted vs. 1 2 the normalized distance y, while the output powers at f and f in db 1 3 4 relative to the input power at f are plotted vs. the same normalized distance y Figure 3.1 shows the case for A (0) = A (O) - 0.0225, f = 1.01 f, 1 2 2 1 C = C = C = 0.05, Z = Z Z O, d O, b = 0076, 1 2 3 4 5 6 7 1 n n (n = 1,2,3,4,5,6,7), op/n = 0o Figure 3.2 shows the case for f = 1.05 f and all other parameters the same as in Fig. 3.1. Figure 3,3 shows the case for f 1.1 f and other parameters as before. 2 1 It is seen that in the linear portion of' the output (up to y ~ 4) the intermodulat'ion output at f and f is about 15 db lower 12, 3 4 than the fundamental output. However for larger values of y when the 1 tube is in saturation or above saturation the intermodulation components get relatively higher. This is of course due to the increased nonlinearity at saturation, In Fig,,ol the maximum output of the single frequency input is shown for reference, It is seen that the power output

-1950 24 f3 C =C2 =C3=C4=0.05, Zh= 0. I Zf 46 A, (O) = A2(O) = 0.0225 f4 f2/f, = 1.0, b=0.076, d=O f3= 2f2- fl =.02 f, f4=2fl -f f =0. 99 fl 42 // 16 //, 38 1 12 MAXIMUM GAIN OF ONE o FREQUENCY INPUT AT y-6.4 34 / — 8 I// < 26 / 0 (.9a 22 -4 w R // ]8 -8 14 / SOLIDzCURVES USE | I0 / -16 SOLID CURVES USE LEFT HAND SCALE 6 DASHED CURVES USE -20 RIGHT HAND SCALE 2 / -24 O 1 2 3 4 5 6 7 8 YI FIG. 3.1 GAIN VS. NORMALIZED DISTANCE.

-2048 -- 28 Cl = C2 =C3=C4 = 0.05 Zh:O. IZf -3 24 44 f2.4 A, (0) = A2(0) =0.0225 b= 0.076, d=O, f2/fl =1.05 40 f3= 1.05 fl, f4=0.95 fl 20 24/ /. // / /RIGHT HAND SCALE 0 O -20 O 1 24 4 5 6 7 8 z 12 0.. DASHED CURVE USE 0 -20 FIG. 15.2 GAIN VS. NORMALIZED DISTANCE.

-2148 28 C, = C2=C3=C4 = 0.05 Zh =O.I Zf 24 44 -f A, (0)= A2(0) = 0.0225 b = 0.76, d=O, f2/f, =.I f=.2f,f4= 0.9f, 20 40 20 36 16 32 12 I — -24 4 Z // I I IlI 20 0 W 16 — 4 oW z 12 — 2 0 -- -2 0 2 3 4 5 6 7 8

-2246 16 DASHED CURVES USE / f4 42 RIGHT HAND SCALE /12 SOLID CURVES USE / / LEFT HAND SCALE / / 38 f3f 8 C, =C2 =C3= C4 = 0.05 A,(O) = A2(O) = v/Y X 0.0225 / ~34 ~f3= 2 f2-fl= 1.2 f f4= 2fl -f2 = 0'9fl / ff2/fI, = I /.1 30 O. Zh IZf 0 < 22I~~~~~ t8 - 0 // z 6 / / -24 -2 1 -32 2 3 4 5 6 7 8 6 - -24 2.-24 YI FIG. 3.4 GAIN VS. NORMALIZED DISTANCE.

-23 - of the two-frequency input has been reduced by about 4 to 4,5 db from the single frequency input, Also the saturation point occurs at a shorter distance. Figure 3.4 shows the case in which the input power of each signal is increased by 10 db. It is seen that the saturation point occurs at a shorter distance (y ~ 4.5) compared to Fig. 353 (y 5.5). Also the 1 1 output at f and f of Fig. 3.4 is 10 db higher than the output of 1 2 Fig. 3.3 up to y = 3.25 above which the output of Fig, 3.4 starts to saturate. It is noticed that the output at f and f (and corre1 2 spondingly the output at other frequencies) is not the same. This is due to the assumption that C = C = C = C. Since, practically, the 1 2 3 4 output does not change with frequency in this narrow band, it seems more realistic to assume some varaiation of C with frequency such that the output at f and f will be the same for equal input power and 1 2 small frequency difference. 3.4 Future Work, Additional results will be obtained for all interesting values of the input parameters and operating condition of the tube. Also the equations presented in the first quarterly progress report have been programmed and solutions will be obtained in the near future. Experimental work on an X-band medium power tube is now going on in order to correlate the theoretical results with the experimental ones. 4. Study of a D-c Pumped Quadrupole Amplifier (C. Yeh, B. Ho) 401 Introduction. In Quarterly Progress Report No. 3 an anomolous gain possibility was reported which occurs when a cyclotron wave is coupled to a synchronous wave of the same kinetic power parity.

Such a coupling mode was previously classified as a passire coupling in which no power gain can be expected. However, in certain types of circuit structures, namely, the staggered and twisted quadrupole structures, high gain is possible under the condition of strong pumpings It was suggested that an explanation of this mechanism be theorized and that some experimental evidence be found to support it, This period has been devoted to these efforts. 4.2 Kinetic Power Relations in Cyclotron Synchronous Mode Interaction. Let us focus our attention to a twisted quadrupole structure" For the coupling between cyclotron and synchronous waves, q = 1c/2O Conditional coupling between the fast cyclotron wave A and -the positive kinetic power synchronous wave A occurs when 2V /a2 > C2 The component 4 p c waves are found to be -Ja e-(P /2) - e(2/4) - M]Z -J[e -( C/2) + J(c/4)-M]z a = f e + f e 1 11 12 -J[$e-(Pc/2) + 4(i2/4) + M]z J[Pe-(Pc/2) - J(pc/4)+M]z + f e + f 13 14 (40X) and -j[~e+(Dc/2) + TD/4j- M]iz -Ji[Pe+(c/2) - (72/4) -MIz a = f e + e 4 41 42 -J[e + (c/2) fi2/4)+ M]z -jie+( /2) ~(/)M]z + f e + f e 43 44 (4e2)

-25where Vp c 2 u cia 0 0 The wo- plot of these component waves under strong piumping condition M > PC /4 is shown in Fig~ 4t1. Notice that the phase of the fast cyclotron component waves f 11 and f have a value of - (1/2)c at X = 0, while f is on its left 12 14 at 1o2,c and f is on its right at 0.2 3co If z is increased, f c 13 c11 and f remain unchanged while f is shifting toward the lefto 12 14 Similarly the synchronous component waves f, f are stationary at 41 42 f /2, and f, f are shifting toward the left and right respectively. c 43 44 For a lossless system, the kinetic power carried in each mode is separately conserved, that is to say, la j2 = constant and la 12 1 4 constant. From the c-X diagram, it can be seen that la12 = 12+ If21- If 31 2+ If 1412 (43) la41 2 [f4ll - tt21 + I1 2~ _f 2 f441 2* 2 (4~4) 4 41 42 43 44 The Chu kinetic power theorem states that P = a I + la I = r12+ 2 f1 2 -jf1 j2 q f 2 - jf j2 - -If!2+ If I2 -! I 12 ifl s + t_If + 4 - 41 42: 43 44 = constant. (4.5)

-26I~~~~~~~~~~~~~~~~~8 H Co_ N~ N Al 4-~~~~~~~~~~~~~~~~0 N 0 ~~ d o~ N 0 a ~ a~~~~ oCM~~~~~~~~~ H i (~

-27It can be seen from Eqs. 41l and 4~2 that the component waves f, f, ff, f have a pure imaginary propagation constant, and thus 13 14 43 44 their amplitudes are constant and so are the squares of the amplitudes0 The power theorem is then simply If 12+ If I12 f I12 If 1 = constant 1 (4~6) 11 12 41 42 where f, f are decaying waves, and f, f are growing waves. 11 41 12 42 Therefore, Eq. 4.6 shows that under a lossless system, the rate of growth of f, f waves is equal to the rate of decaying of f, f e Further12 42 11 41 more, it indicates that f, f carry positive kinetic power, while f 11 12 41 f carry negative kinetic power. In other words, the so-called positive 42 synchronous wave now carries a net negative kinetic power which can be coupled to the cyclotron wave to produce amplification. Similar reasoning applies to the slow cyclotron-to-negative synchronous wave coupling. Since the signal input and output are coupled through the use of Cuccia couplers which are isolated from the pump field region, the bandwidth of the amplifier is limited by that of the input and output couplers. However, a possible wideband operation can be obtained by using a quadrifilar helix as the input and output coupler, and at the same time as the pump field structure, such as is used by Mao and Siegman' in their simultaneous r-f and d-c coupled cyclotron amplifier. lo Mao, So and Siegmanr A. E., V'Cyclotron Wave Amplification Using Simultaneous R. F. Coupling and D. C. -Pumping", International Congress on Microwave Tubes, ppo 268-276; 1960o

4.3 Experimental Evidence of the High Gain Transverse-Wave Amplifiers. Thus far, there have been two papers reporting the "unexpected amplifying phenomena" without due explanation"l2. In the paper by Saito, Kenmoku and Matsuoka, the observed high gain was the result of reducing the drift voltage to a quarter of its normal operating value. This reduction in drift voltage corresponds to a reduction of the drift velocity to one half of the normal operating value. This operating condition corresponds to the condition for cyclotron-to-synchronous wave coupling at Pq =(1/2).e If this pump voltage remains unchanged under this condition (which is assumed, there is no other place to indicate otherwise), this is precisely the necessary strong pump field condition for the system to exhibit gain, Mao and Seigrnan reported a similar gain increase in which they have puzzled that according to the operating condition it would indicate a cyclotron-to-synchronous wave coupling from which exponential gain is not possible. It is our belief that strong pump field will shift the phase of the component waves in this type of coupling to make the exponential gain possible. 5. General Conclusions (C. Yeh) The experimental low-frequency model of a frequency multiplier with feedback has been constructed. The d-c operating characteristics of this device have been checked and are found to be satisfactory. The device is now awaiting final r-f testing of its workability and efficiency. 2. Saito, S., Kenmoku, M. and Matsuoka, T., "D. C.-Pumped CyclotronBeam Tubes Using Quadrifilar Helix"' International Congress on Microwave Tubes, pp. 244-248; 1962.

-29Difficulties encountered in brazing a tungsten helix for operation at 30 Gc into a smooth-bore BeO tube have not been fully resolved. Several other alternative methods are being tried. A general analysis is given to describe the nonlinear operation of the traveling-wave amplifier with multifrequency input. Computer results for the system are discussed, The explanation of the anomalous gain phenomena in terms of kinetic power theorem seems reasonable. Strong evidence of the existence of such coupling is found in two entirely unrelated experiments. It is planned to design a tube which will verify this phe-nmena more directly in the near future.

DISTRIBUTION LIST No. Copies Agency 3 Chief, Bureau of Ships, Department of the Navy, Washington 25, D. C., Attn: Code 681A1D 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 RAAV-61 1 Chief, Bureau of Naval Weapons, Department of the Navy, Washington 25, D. C., Attn: Code RMGA-ll1 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

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