ORNL/Sub-75/49438/2
BP n ^TfFl varian mw 3 % h
NOTICE
PORTIONS OF TV'!? HFTL0'' SHE M.I.HGIB'.E. IF lT~~rvnrT;irri*"rr?.--p. tf.3 iwaito'jia 1 co1;;/ -o psrrnii the broadest possible avail- ability. FINAL REPORT MILLIMETER WAVE STUDY PROGRAM u by H.R. JORY, E.L. LIEN and R.S. SYMONS
ft, 0
{I Order No. Y-12 11Y-49438V j November 1975
report prepared by
Varian Associates Palo Alto Microwave Tube Division 611 Hansen Way o Palo Alto, California 94303
under subcontract number 11Y-49438V ; for ' ' r>4 OAK RIDGE NATIONAL LABORATORY * ° Oak Ridge, Tennessee 37830 operated by UNION CARBIDE CORPORATION for the nrsTKIUUTiON 01? 'ijl;.;?!---''--^-'.^-^''' UNLLMIT DEPARTMENTS ENERGY FINAL REPORT
MILLIMETER WAVE STUDY PROGRAM by H. R. Jory, E. L Lien and R, S. Symons
- NOTICE- Urn report Mas piepated as an account, of worl; sponsored by die Untied Stales Government. Neither Die United State* not the United States Pepastment of l.nergy, tiar any of1 their employees, nnr any of then contractors. subcontractor or their employees, makes any warranty, express or implied, or assumes any legal liability oi responsibility for tlie accuracy, completeness of usefulness of any information, apparatus, product or process disclovd. or represents that its use would not mUmfe pnvately owned npjits.
Order No, Y-12UY-49438V November 1975 « 0
" report prepared by
Varian Associates Palo Alto Microwave Tube Division 611 Hansen Way Palo Alto, California 94303 K< under subcontract number 11Y-49438V for OAK RIDGE NATIONAL LABORATORY Oak.RitJge-J'ennessee 37830 c.-..operatedjby U N 10N CARBiDE^G0RP0RATI0N, o
Section - 9 " , ' „ Page No.
1. INTRODUCTION...... o . ... . 1-1
' //
2. WAVEGUIDE DESIGN . . 0 " . . . . , . 2-1,,
3. . WINDOW DESIGN 3-1
4. LINEAR BEAM AMPLIFIERS >< . 4-1 > i 4.1 Introduction ,4-1 4.2 TWT Amplifiers •<....".•;.„ 4-3 4.3 Harmonic Amplifiers 0 v. ; 5. PERIODIC BEAM DEVICES ...;...... 5-1 o 5.1 Introduction ...... 5-1 5.2 Ubitron ' .. 5-1 5.3 Periodic Beam TM Interaction 5-4
6. CYCLOTRON RESONANCE DEVICES . . q. 4 6-1
6.1 General Description of'Device Alternatives . . . 6-1 6.2 Analysis of Cavi,ty Losses in Cyclotron Resonance Devices 6-12 < 6.3 Gyrotron Giui Design . - ,6-16 6.4 Axis Encircling Beam . 6-22 f6t. 5 Gyrotron Interaction Theory ' • " 6-33
6.5.1 Introduction and Summary , e , . = 6-33 6.5.2 Review of Russian-Theoretical Work . . . 6-35 • 6.5.3 Analogies Between Klystrons and Gyroklystr.ons 6-42 6.5.4 Determination of Performance Parameters Using Numerical Methods 6-48 6.5.5 Ballistic Trajectory Analysis Code. . . . 6-58
7. CHOICE OF APPROACH ..." 7-1 o 8. DESIGN CALCULATIONS FOR THE PREFERRED APPROACH 8-1 v. 8.1 General Description .J 8-1 8.2 Microwave Circuit Design . 8-3 8.3 Electron Gun Design. ~ ...... 8-10 8.4 Foe using'Solenoid 8-11 £r .8.5 Collector and Water Cooling Design ...... t8-12 8 .6 Waveguides and Windows xr- „. • 8-12 t? • (x sr ° 8.7 Summary of Design Parameters ...... , 8-13 o iii
TABLE OF CONTENTS (cont'd)
Section « Page No.
9. CONCLUSIONS 9-1
10. REFERENCES 4 ,,10-1 „ O• ' M , • , APPENDIX A
APPENDIX B , „ 'll LIST OF ILLUSTRATIONS
Page No.
Attenuation Due to Copper Losses in Circular Waveguides o at 120 GHz Assuming.a Skin Resistance of 0.09 ohms. 2-4
Measureo d Window-Loss as a Functio11 n of Transmitter Power. '" 3-5 f) I _ Dimensions of a Disc-Loaded Waveguide Designed for Forward Wave, Operation at, S-band at a Voltage Level of 200 kV. 4-5"
W-/S Diagram for the Disc-Loaded Waveguide Circuit f/ • Described in Figure 3. ' t ' 4-6 0 & •> , l' ' " ^ • Measured Impedance for'the Disc-Loaded Waveguide at a Radius Equal to 0.84 Times the Beam Hole Radius. " 4-7
Modified Applegate Diagram Showing'the Phase of>sthe Various Disc-Electrons vs ^Distance Along the Tube." '' ' '' "ij -- i u Fundamental Component of Beam Current Along the Tube ' " Together with Gap Voltages. '' is 4-10 (j ' " As 'V, II . , Average Valuecof Radial Factor of Beam Coupling Coefficient .for Uniform Density Beam ^Partially Filling Gap. « " 4-12
Phase" vs Distance for Electrons in Output Section of Tapered ^Impedance TWT with.Output Power of'110 kW. . \ 4-17 '
RF Beam Current and Cavity Voltages of Tapered Impedance • ,t n-i^T7rn'x vv j. ;WJ.U 1 illifui 1-Titv»vr v_/UlpUtrtii^nf. " fr^jlv"? oi o
Phase vs Distance with Impedance and Velocity Tapers and 207 kW Output of-20
a ' " Beam Currents ancrCavity Voltages with Impedance.and Velocity Tapers and 207 kW Output... ° 4-21 0 0 Output Circuit with Tapered Impedance. „ » <=> 0-4-22 o Modified Applegate Diagram Showing Performance on,a Tube >. , 0 with Cavities Spaced and Tuned as Shown in Table VTEP « 4-27 _ , , ' - <1 , • n Development of the Fundamental, Second Harmonic and Fourth Q ^ Harmonic Components of Current; «, : 4-29 a * v. 3 0 « , a tP Kf-Ps & • a • LIST OF ILLUSTRATIONS (cont'd)
Figure Page No.
16 Schematic Drawing of V-bancl TJbitron Amplifier 5-3 is « 17 Gyrotron Configuration (after Zaytsev). " „ - 6-3
" O te „ 18 " Cross Section for 02I Gyrotron. 6-5 i'l o H 19 Cross Section for TE „ Gvr.otron. w •6-7 ^ nn231 " <"-' „ i) 20 Drawing of Experimental Gyrotron and"the Axial Distribution » of the Magnetic Field (after Kisel1) . " 6-10 i, « v!l Raclial Variation of E . for TE „ Cylindrical Cavity. cp " n,\l, 1 ^ 6-11 ' a \\ CI - 252±; Figure of Meriffor TE- • Modes. ' - 0,m,l 6-15 CI I Figure of Merit for TE , , Modes. ' „ 6-17 II 24 Posisible'fElectron'Orbits in Hollow Beams Useful in 5ssi. » U 6-23 TE „ jDevices, ° Ml i'. o 25 Beam Envelope with Magnetic Reversal. 6-25 26 Coordinate System for Electron off Axis. 6-27
Electric Field for Optimum Efficiency for Osc illatbr"Analyzed 6-40 tby Rapoport, et al. o < -u
31 Model Showing Power Flow in the Amplifier. 6-49
32 tt 100 kW, 120 GHz Gyroklystron. 0 S -
33 Electron Energy vs Time with 'Synchronous Magnetic Field. 8-7 '1 j o 34 Electron Energy vs Time with Magnetic Field\Set for 4.5% Slip. 8-8
• o '." -
9 vi
V ft * LIST OF TABLES u a TABLE PAGE NO.
Diameter/Wavelength'for TEnm Mode Cutoffs in( Cylindrical Guide. 2T2
II Diameter/Wavelength for TM^ Mode Cutoffs" in Cylindrical Guide. m Data for Various Commercially Available Window Materials at X-Band. VI Window Test Results Summarized. 3-4 V Voltage-Dependent Parameters of One-Megawatt Beams. 3-11 VI Brillouin Focusing Field and Area Convergence for Various One-Megawatt Beams of Different Energy and Size.,, (\ 4-13 •o VII Parameters of Selected One-Megawatt Beams. 4-15 . " XI Parallel and Peroendicular Energy for Rotating Beam, 6-33 i, XII Tentative Design Parameters. 8-14 u a o 'if \V v^5 0 vii 1. INTRODUCTION rW:- The purpose of this program was to study the various approaches to \ " ' " - II building an amplified to produce 100 kW or more cw power at 120 .GHz to decide on an optimum approach, and,to perform design calculations. The stuc .y has led Q to the conclusion that a cyclotron resonance amplifier (gyrotron) is the optimum approach for 100 kwland that this type of interaction offers the best possibilities o 1 ;i • P for going to still"higher power levels. ! This report includes discussion in depth of all"the topics and approaches ! . ° 1 considered in the study. Sections 2 and 3 deal with output waveguide and window ' | " c • >. ' " O "I •• o designs. The fourth section discusses linear-beam amplifiers. The fifth section . o. It „ . II .considers periodic-beam (other than cyclotron resonance) devices. !! I ,. . 0 si • ii - i Some readers may wish to bypass"these early sections of the report and i' I iv begin with Section 6 which considers cyclotron resonance devices in detail, or • 1 Section 7 which summarizes the. arguments leading to the choice of the gyrotron r as the optimum approach= ". Section "8 contain' s design calculationn s for a 10ii 0 kW \ (I gyrotron amplifier based on the background and theory presented in Section 6. , 6 (j e •fi c 4> 4 l-l • 2. WAVEGUIDE DESIGN ' ; ' o • • v • / " A single-mode, rectangular TE waveguide, 0.080 inch wide by 0.040 inch high, has an attenuation of about 4.3 dB/meter at 120 GHz1 assuming ri skin " 'n .• • • resistance of 0.. 11 ohms per square for copper. This is more realistic than the « w ^ dc based value of 0.09, although when surfaces are not htghly polished (electro- polished for example) skin resistance can easily reach double the dc base d 11 value. Even if this attenuation were acceptable, the power density on the waveguidu e surface 2 ' " at 100 kW would be a 1.6 kW/cm which would require heroic cooling measures, " and the electric field would be about 70,000 V/cm. For"this application, 'therefore, it would seem that there is little choice other than a multimode waveguide. Once thisc.decisionQ has been made, it follows that because of mechanical considerationQs and because of the potential advantages of the TEQm family of modes, we should concentrate our attention on circular guides. ° v. v , 15 *" !*J Tables I and II show the diameters normalized to wavelength at which high >" • - 0 order TE and TM modes begin to propagate. We can see that, for a waveguide nm nm & • ' ° having a diameter of 3 A, more than 40 modes can propagate. Foi' a diameter of 2 A the situation is considerabloy simpler, and only nine modes propagate. For the problem at hand, higher mode power delivered to the ELMO cavity will probably " „ be just as effective for heating electrons as lower mode power. However, the O ? t. " • higher modes will have higher losses in the waveguide and very high losses indeed jt f. .. 'V near their respective cutoff frequencies. It is well known that if a suitable coupling irregularity exists, the high (J impedance of a mode near cutoff enhances,the energy transferred to this mode. Therefore, from a loss standpoint alone, we should concern ourselves with choosing a waveguide size for which the operating frequency lies between cutoffs rather than , " o on one. We should also concern ourselves that no mode, once we permic t it to propagate toward the load, is cut off again so many length-related resonances occur to give enhanced mode conversion, loss and high fields. Should it be necessary to shrink the waveguide size it would be well to provide loss, 'at the point of maximum " - , - ' iJ .i waveguide?size, for modes which will subsequently become non-propagating. v. ' ' "A" c . II TABLE! I 0 ' i, It. r, Ditiinstsi^ — : rr for TE Mode Cutoffs in Cylindrical Guide Wavelength nm jj n 0 1 «2 " • 4 5 6 7 •> m \ I3 . !!« •" •o 1 1.220 0.586 ' 0.972 It 337 1. 692 * 2.042 ' 2.387 2.7,30 " O U h .2 2.233 - 1.316 2.135 2.551 2.955 3.349 3.753 J(4.116 f tt / o . c- <) U fj 3 3.238 2.717 3.173 3. 612 4. 037 4.452 ' In! O - r, i • > it 4 4.241 3.726 4. 192 ° ' TABLE n 0 ' .. yff - Di ameter for ,TM Mode Cutoffs in Cylindrical Guide Wavelength '' nm n o 0 1 2 3 5 6 7 m Nv • o C Cf t. 635 2. 031 2.415 1. 1 " 0.766 1. 220 " 2.792 c3.163 3.529 ® " o 0 () v 2 l.(757 2.233 2.679 3. 107 °3.522 3.928 ' 4.'326 <4. 718 1 3 2.755 3.238 3ft699 4.143 4. 575 G \ "11 • ft i % 3.754 4.241 4.710 0 o , 0 " £ o& O • jr G> 2-2 o Sr It is also well known, that certain discontinuities excite mode families, and even some m©des within a family," preferentially. For 'example, discontinuities that are symmetrical about the center line of the guide will convert the TE "to,TE., modes-where m ;;is odd. Assymetrical discontinuities will excite modes for which m is even. The telephone company has made good use of the fact that" it is difficult to convert TE„ modes to TE or TM modes where n = 0, and it is easy to 0m nm nm % :'build mode filters and lossy elements which keep .the TE„ modes, pure while lcWer- ' // • .. • 0m •• <• i; ing the impedance on TE and TM modes. These techniques could be used f/ <> ? if needed on this program. " ' <> a " •-• 1 ' n , ' U f •Returning to Tables I and(ll, 'we see that if r.ve use a waveguide diameter' of about • v- i 1. 9 X (0.1S7 inch) we"will be relatively far from all cutoffs. Figure ! shows the attenuation for some modes which will-'propag'ate in such a waveguide at 120 GHz. The curves are calculated,for,a skin resistance of 0.09 ohms1. At the value of A/X 'of 1..Q'* the attenuation for either, theiiTE or TE mode'is about 0. o6('dB/meter0. For „ " K) 01 • 11 -i . _ O . 100 kw in°the guide, .this becomes 129 W/cm or SGW/cm"!. This seems reasonable $ „ '' u « . " a •<• » - •••• ' from both a cooling and total loss standpoint'' . The ..electritJ c fiel... d will bQe aboui ir t * „• - n,* " 0 eK, a p - •• „ » » " 30,000 v/cm peak for the' TE.. mo'ae'and 34,000 v/cm peak for,the TE mode. " » , • 11 '•• o '' <•> • • 01% • The voltage for fundamental mode multipactor2 has been calculated for, this guide1 '" " • i n ° „ " '/ ft size', and for a secondary electron emitted at the instant the field changes sign, it ' * " ^ 6 . ' uf3 ' v „ - is 23 X 10, V. Because the mode voltage for the TE mode is about 9,00Q v peak, p ° '* ' . • ' * " 111 : '"' " j; ...... ; . ti .(, there is> even more clearance than in typical lowexv5requency; 100 rkW tubes using § 0 •• normal TE.. rectangular waveguide; This is because-the miiltipactor voltage scales' < o • , » ,, f as (d/A)". r " , v o tj " •/ (S . /. ' - <•- s> o V To summarize the preceding discussion: 0 % * v, " " . t*i O 0 • O » 5.1 ° , „ - •" 1) The loss, cooling problems.and fields associated with a fundamental '' ' Vi « ' o\ , o 0 a mode waveguide seem unreasonably high.," ,, . s • ,•> e. " .. • O6 ' o "... ., 1 " y r' 'o I, " • e ° u • 2) A circular wayeguide having a d/\'= 1.9 (0.187 inch for 120 GHz) will H 0 have losses of 0.56 ciB/meter for,;either the-TE or TEQ1 modes. „ 3. 75 3. 50 TJ TE-LI® I 3. 25 * TE i Y 01 C O • 00 CO 2. 75 dM) \ C a) 2. 50 a 2. 25 \ v. = c }• CD £> 2. 00 • r-l \ \ O 'atu \ 1. 75 \\ 50 1. \\ to > I § 1. 25 \ 0) \ £ 00 < 1. 0. 75 \ ii 0. 50 i: 0. 25 rv 0 0.4 0.8 1.2r 1.6 2.0 2.4 2.8 3.2 3.6 4.0 Diameter Wavelength Figure 1. Attenuation Due to Copper Losses in Circular Waveguides at 120 GHz Assuming a Skin Resistance; of 0.09 ohms. e - Losses for these modes will be 129 W/cm of length or 86 W/cm" •'I ' ' of waveguide surface area. Eight other modes can propagate. The operating wavelength is not near the cutoff of any mode in order to minimizeO mode conversio' n problems. Once the waveguide size is increased (as will be necessary at a window for example) either loading for higher order modes should si 0 i be provided at the point of maximum waveguide size or else the wave- guide size should increase continuously toward the load so that no high-impedance, length-related resonances exist to absorb power. The. TE^ mode should be seriously considered because of the ease of mode filtering and of damping other modes. For d/,\ = 1. 9, the TE^. is the only TE mode which will propagate. 01 f°irr~ Multipactor is no problem. 3., WINDOW DESIGN f V ( i , An approximate calculation can be made of the power which will crack a vacuum window.Jj-For surface cooling, the window Breaks because the cooled surface is put in tension by the average thermal expansion which results from dielectric, heating. For edge cooling, the power that can be put through a window is sub- 0 stantially reduced, frequently by as much as a factor oE ten, because of the larger temperature differences that exist oyer the longer heat-flow path. In Appendix I an equation is derived for the limit on the square, of the electric field that can be applied to a face-cooled, thin ceramic window. It always predicts larger limits i • than, in fact, occur. This is due in part to the fact that, in a practical window, there is always some radial heat flow. The surface conditions such as microcracks, nicks, etc., also exert an enormous influence on the rupture stress. The equation is quite useful for,scaling, however. It is ,;< -"oJ " " <., '^S E2 = 12,6 f1 . 2 \x k e" M / (1w) OJ e t 1 ' o it in which E is the electric field in peak v/cm, y. is Poisson's ratio (0. 25 for isotropic materials), a; is the angular frequency in radians per second, t is the thickness of the window, e is permitivityJ of free space in Farads/cm, S is the modulus of o - max t rupture and M is Young's modulus in identical units; I<9 is the thermal'conductivity in cal/sec cm c C, k is the expansion coefficient in parts/part/° C and e" is the loss factor. 0 The quantity within the parentheses in the equation can be considered a figure of merit for the various dielectric materials which might be used for high-power windows. Table HI gives the various parameters for some commercially available alumina and beryllia ceramics and alsot' for 0° cut sapphii-e at X-band. For 0° cut sapphire it should be noted that k , k0, and e" are the same in all radial directions in the plane of the window so the anisotropy is not important here, but j.t is not' 0.25, so the comparison is not strictly valid. Examination of the data in Table in shows 3-1 TABLE III Data for Various Commercially Available Window Materials at X-Band " o Tensile Young's Coefficient Thermal Loss. Figure Strength Modulus of Expansion Conductivity Factor of Merit psi psi 1/°C cal/sec. cm. °C C3 Coors r* AD°995 38 x 103 °54 x 10 7.1 x 10"° 0. 062 0.002 3072 99.5% A12°3 Coors n BD 995-2 20 x 103 51 x 10 6.4 x 10" 0.48 0.003 9S03 99.5% BeO Coors ^ f* ~ Vistal 30 x 103 57 x 10 , 6.5 x 10" - 0.068 0.001 5506 Trans luscent A12°3 G.E. Lucalox 40 x 103 o 57 x 106 6.5 x 10"° 0. 06 0.0025 2617 - „ Transluscent A1 2°3: Linde 3 6 Sapphire 58 x 10 56 x 10 ; 5.8 x 10~6 - 0. 06 0.0028 3S26 that, (because of its high thermal conductivity:, beryllia has the highest figure of merit'even though its strength is less and its loss factor is higher. 1 f / Data above X-band on ceramics are quite inadequate. Coors gives some t data at 50 GHz which;are disturbing in that for AD 94 the loss factor is double that at 10 GHz and for AD 96 it is more than ten times as great. No data at all are " 0 given for the high-purity aluminas or bcryllias which at 10 GHz have the lowest losses. We have contacted Coors, but no additional datna exist. It will probably be necessary to make loss measurements. • L ' .. ' f In 1966 and 1967 the Eimac Division of Varian had a contract to build a 1 MW X-band window. The succcssful window was an edge-cooled, sapphire disc. 3 Table IV, which is taken from the final report on this contract, gives data on the various windows tested. s*Figure:<2 shows measured loss as a function of transmitted power., Probably the only significance of these data is that both BeO and sapphire are good candidates for high power windows, and that there have been substantial a f \ increases in the strength of BeO between 1966 and the present time. Table HI is based upon 1972 data. The data in Figure 2 also seem to indicate that e" for sapphire is lower than shown in Table m. if the loss factor does not increase substantially for BeO or sapphire from the 10 GHz values, our 1 MW X-band experience indicates that a 100 kW, .1'20 GHz window should be possible. Even if the loss factor increases somewhat, there is still a possibility we can face-cool the wiiklow with high-velocity gas jets. .Calculations" • Q indicate that this process may improve the power capacity of the window by as much as a factor of ten. , A factor of more than four has been demonstrated on a UHF klystron. This demonstration was limited by the available power. ' • We wi)l now examine possible physical configurations for the window. If we <:' " ' 2 multiply both sides of equation (1) by h , the square of the waveguide height, we see that the left si'de of the equation is the square of the mode voltage in the waveguide and the right side contains (h/t) so that, for a constant ratio of window height to thick- ness and constant waveguide impedance, the power capacity will scale as l/cj. At TABLE IV // Win'dow- Test Results Summarized ^ ' TRANSITION VOLUMETRIC TRANSMITTED MATERIAL TYPE THICK DIAM RADIUS % LOSS* DISSIPATION RATE POWER COMMENTS W/fctV/cm3 kW 99.5% BeO Thick Disc Long 0.238 1.316 0.187 0.254 1.22 600 - 700 Window failed when No. 1 Pill'Bo;: power was raised from Undersize 600-700 KW - vertical ciack. 99.5% BeO Thick Disc Long 0.238 1.316 0.187 0.22 1.06 810 Window failed while No. 2 Pill-Box running at 810 kW — Undersize diagonal crack. 99.5% BeO Thin Disc Short 0.090 1.680 0.063 0.296 2.31 400 - 463 Window failed when No. 1 Pill-Box power was raised from Oversize 400-463 UV 7- diagonal crack. 99.5% BeO Thin Disc Short 0.090 1.680 0.063 0.27 2.1 400 Window failed after No. 2 Pill-Box 2 minutes at 400 kW - Oversize diagonal crack. . 99.5% BeO Thin Disc Short 07065 1.316 0.187 0.145 2.5 262 Appears to have cracked Pill-Box twice — once at 262 kW and^ Undersize again at 320 kW _ CO Sapphire Thin Disc Short 0.050 1.603 0.187 0.116 1.78 1050 Failed after tunning at 'tSO~* cut Pill-Box ~ 1.05 megawatts for 7 min. - Normal Size "V'-shaped crack. SapphireA! Thin Disc Short 0.045 1.316 0.187 0.07 1.78 400 Failed at 400 kW after 60° cut Pill-Box running jpprov. 30 sec. — Urvdcrsize O jagged ciack nearly o horizontal. ^ Sapphire Thin Disc Short 0.050 1.316 0.187 0.095 2.16 1000 DID NOT FAIL. Continous 0° cut Pill-Box tunning time at 1.0 MW = Undersize 37 mi'i. Total time at 1.0 MW = 39 mill. "(Corrected) calculated copper losses subtracted. ? 0 Cu it Transmitted Power (kW) Figure 2l\Measured Window Loss as a Function of Transmitted Power V 3-5 i! 120 GHz, there is a practical problem, however. That is,' a window that is 0.045 inch thick at X-band would scale to about 0.004 inch at 120 GHz. .Even if it were vacuum tight at this thickness, there might well be problems in metallizing and brazing it to the waveguide. In addition, %iere will always be some energetic electrons around in a vacuum with high rf"fields, and total dielectric strength cannot be ignored. An alternative is to expand the waveguide into a horn and use an oversized I window at the ape.rture of the horn. In this way, the same value of h/t can be used with a more reasonable value of t. In this case, the window thickness will have to be an integral number of half-wavelengths. The price of increasing the size is that there will be an enormous number of higher-order modes which can exist/in the window. On the air side of the window, however, it should be possible to load these resonances by leaving a gap at the edge of the window. Only the TE^ modes,"' which areocut off in the waveguide beyond the window, will have high Qs, and it may be''possible to proportion the window so the operating frequency lies between two of these modes. While the prospects for a window which will withstand the power seem t' > reasonable, it will require substantial development, and it would be wise to also follow H some alternate approach, such as a vacuum valve for the waveguide. e In conclusion: ". 0 1) Sapphire and beryllia appear to be the best window materials, however, >5 V !? data at 120 GHz are unavailable. 0Some of the newer high-purity aluminas should-also be examined. 2) A disc window, an integral number of half - wa vel e ngth s thick, across the mouth of a horn seems to offer attractive possibilities. Higher " order modes can be loaded by a gap in the waveguide on the air side oc of the window. o 3) A windowless design should be pursued in parallel, in case the window problem proves too difficult or becomes the limiting item. o // ^ 3-6 " o • „ 4. LINEAR BEAM AMPLIFIERS r v. -V. • - ft ' 4. h INTRODUCTION 1/ TKe major problem iii,building a high power tube for operation at very •high frequencies is'circuit dissipation. First of all, the skin resistance of the circuit material varies as the square root of the frequency. Second, o for direct wavelength size scaling, while maintaining the operating voltage, i> the magnetic field at the walls and the electric field in the interaction regions will increase directly as the frequency. This occurs because the length of the interaction region is proportional to the wavelength. These two effects cause a 5/2 power dependence of circuit power density upon frequency. 1 f " - .. " • . Iii addition, the situation is exacerbated by the fact that,limitations u r O ^ on possible cathode current density and temperature make it necessary to limit beam current in order to form a beam of suitably small diameter. Because of this the operatini. g ' voltage canno1t even be held constant'S, but instead it must be increased to achieve the necessary beam power. One therefore finds "hirasel(( f in the position of applying 5/2 power, power-density scaling to a design that, before scaling,"was less than optimum on a power-density basis. h There are two ways out,of this dilemma. rOn'e is to use cavity or circuit modes which tend to increase the electric field in the region of the beam while maintaining low magnetic field at the wall. The second is to consider, and j 0 attempt to improve upon, only those designs which stretch the interaction length to a°maximum, thus maintaining the electrons in interaction with the lowest possible field for the longest possible distance. „ . j In this study we have attempted to evaluate seyeral kinds of tubes, which possess one or both of the above features. " " ' ' - „ I • - 4-1 e- For the case of linear-beam tubes'; traveling-wave tubes and extended- interaction,ldystrons represent the types with the longest interaction lengths. • ({ (Simple klystrons have impossibly high fields.) Extended-interaction klystrons employ either a standing-wave resonator, with a slow-wave circuit having a reflection at both ends, or a traveling-wave resonator in which power from the output end of the slow-wave interaction circuit is recirculated to thie input end. " In these circuits there must be a large amount of circulating energy if the field is to be relatively constant along the output circuit, and there'will be loss associated with this stored energy. ' } Traveling-wave tubes have a minimum of stored energy, but unfortunately, '' the rf fields build up exponentially along the circuit. At 120 GHz, if the output 0 ° " " circuit impedance is high enough to make circuit fields at the input end of the output circuit sufficient to hold the electron bunches together, then electric and magnetic fields and circuit losses at the output end of the circuit are intolerable. During the program it \vas found that lowering the output circuit impedance enough to overcome the loss problem caused the efficiency to decrease significantly O " due to debunching. A modification, of the output circuit was found, however, which provided adequate bunching with tolerable circuit losses. By tapering the impedance of the output circuit downward by a factor of ten from the sever end to the output end, it was possible to obtain an efficiency of 20% with reasonable circuit losses in computer simulations. (/ While the circuit is complex and the fabrication would-be a prodigiou's task, we believe that this approach could be developed into a tube of substantially greater capability than those currently available at 120 GHz. The next section describes, in chronological order, the design steps which t. Cl led up to the tapered impedance TWT. Extensive use was made of a proprietary Varian large-signal program which uses a disc electron model to predict the a „ <5 - <3> performance of TWTs and Idystrons. Similar'programs have been developed and described by Tien4, Webber3 and Howe6 . " 1) 4-2 tv V ( I' , Another approach which was investigated, though far less thoroughly, \vas^that i) of-a frequency ^multiplier. „It offers the possibility of increasing the size of ttie drift f i tube enough to increase the current-^ubstantially while decreasing the voltage. In such a'device there could be high gain because there would be no danger of feedback between the bunching cavities which would operate at a frequency below the cutofi of the drift tube. A traveling wave output circuit would still be necessary, however, and the most serious objection to this approach is that even at 150 kv proposed for the TWT approach outlined above, the cavity height was only about 0.015 inch. If the it voltage is lowered the pitch would become even less and the circuit would become still more difficult to fabricate. Another problem is that if one achieves lower circuit losses through a higher current and a lower voltage at tire expense of a larger beam and a beam hole'which propagates, many'modes, one approaches oscillation conditions from two directions simultaneously (i.e., larger electronic conductances and increased mode problems). Also, we faced the practical problem that we had no computer program which would predict the buildup of power "-&h a traveling-wave circuit when //a beam bunched at a sub-harmonic of the output frequency was introduced, nor did we have timeVw develop one. For these reasons, work on this approa^h wa's terminated early in the study. The work that was done is summarized in Paragraph 4.3 of this section. ;J a ° 0 4.2 TWT AMPLIFIERS u ' O In a linear-beam TWT the beam diameter is limited firstiby the maximum siz which will, of course, result in the largest circuit, and " ' „ ' ° " « I, W U " ' " ^ .. The required beam power is determined by the efficiency. Initially, it seemed reasonable to assume we would achieve 'a rather low efficiency in the tube because efficiency is adversel« y affected by If rather, poor beam coupling, " .-J resulting from a desire to minimize both beam focusing problems and tolerance u „ „ problems by using a large beam hole. In order to determine more accurately what'efficiency we might expect, we first assumed we could achieve an efficiency of 20% for a 120 kV, 4 A, 1°. 25 mm diameter beam.,, Impedance data and a '<> Brillouin diagram for a disc-loaded waveguide circuit which scaled to give a 1.5 mm diameter beam hole for synchronism at 120 kV and 120 GHz were availabl" e from j! a Watkins-Johnson report7 . These data are shown in Figures 3, 4, .and 5. u " Although we have some, reservations about the size of the drift tube and possible propagation of the TE ; mode, the circuit is fairly close to wh'at might be used. cy j lt The characteristics o„f, severa§ l traveling-wave tubes using-this beam and circuit were calculated usinga Varian proprietary large-signal computer program. a o fi o This program utilizes'a disc-electron model of the beam including relativ- „ ( <1 • istic effects and a three-port (input, output, and beam coupling) network model 1 of the circuit cavities. ^It*" has given excellent agreement with tube performance ' . iJ in the past. While it is the mos,t advanced large-signal program,,available, the Q, electron is represented by a^ rigid disc,' and so we are cautious about« its use hereV; because' of the lar,ge beam and circuit diameters (relative to wavelength) which 0 ' P " ^ we must°use in the tube0underf'consideration. " KJ v ' c n -o " ' &' > „ f " b- v o i Tlie best results of the large-signal calculations were obtained for a 11 % severedpcircuit TWT with an inpuit section of 29 cavities terminated at a sever and "o an output section of 16 cavities. The output circuit had a termination at the output end and a short circuit at the sever end. The short-circuited sever increased the ^ " * „ fields in the output section, reduced the required length, and raised the efficiency.. However, as will be discussed later, this was a mixed blessing because the electric fields became rather high. For this tube the efficiency was 14.6%, 'correspondinug ' to an output power of about 70 kW,. Figure 6 is a modifie° d Applegate 3-4 / 0" 2 fi> A 2.20" o 1.25" \ 0.5.0" 0.24" j-O- a * * a y. C £ •f'l o Figure 3. Dimensions of a Disc-Loaded Waveguide Designed for Forward Wave Operation at S-band at a Voltage Level of 200 kV, u o>. a>a 3 ITM S-i // 7T /2 3T /4 jr 5tt /4 3TT /2 Phase Shift per Section (/3 L) Figure 4. co -fi Diagram for the Disc-Loaded Waveguide Circuit Described in Figure 3. (The mode designations correspond to what we believe is the cavity resonance, when the coupling hole is very small. There is some doubt about whether some of the higher frequency modes should be labeled TM or TE, but the number designation of the 0 and r variations has been determined by probe measuremer 4-6 200 u tn 3 a 'o 100 90 „ a 80 a 70 CO 60 o o 50 a w 40 30 0) o aCD C 20 o +oJ I C4 Qf-Jl 10 0 1/5 2/5 3/5 4/5 /3 L/tr Figure 5. Measured Impedance for the Disc-Loaded Waveguide at a Radius Equal to 0.84 Times the Beam Hole Radius 4-7 V. 0 .40 0.M5 0 .50 0.55 0.60 0.6S 0.70 0 ,7S 0.80 C.SS Z. INCHES Figure 6. Modified Applegate Diagram Showing the Phase of the Various Disc-Electrons vs Distance Along the Tube. 4-8 diagram (showing the phase of the various disc-electrons as a function of distance along the tube relative'to the phase of an electron of dc beam velocity.) Figure 7 is a plot of the fundamental component of beam current along the tube together with gap voltages. The gain of the tube was 25 dB. One disadvantage of this design, in addition to the low-power and the possible TE problem, is the high rf field in the output circuit. In the cavity having the highest field, the voltage was 0.0S times the beam voltage. The gap was 0.3 mm long. The peak field, therefore, was about 300 kV/cm which is equal to the highest field wc have used in X-band, 100 kW 2 klystrons. Two kilowatts were lost in thif ^avity which had an area of only 0.04 cm" 2 giving a power density of 50 kW/cm". This is intolerable and implies at least a 3000° C temperature rise at the cavity wall for radial heat flow through copper to a surface that could bj, e water cooled. Clearly, we had to reduce the circuit fields by a factor of 3 or more and the circuit losses by a factor of 10 or more to arrive / at a practical design. We thought this might further limit efficiency. Because of the previously-mentioned low efficiency and power'and TE ^ drift tube propagation problems, we decided to see if we could raise the beam 6 ~~ power to 10 W, so a 10% efficiency would yield 100 kW. Table V shows, as a function of beam voltage, the beam current for 1 MW, the corresponding perveance, v/c, the relativistic mass correction factor, and the longitudinal and radial wave numbers, k' and k'^, which differ due to relativistic effects8, and are used to cal- O culate longitudinal and radial beam coupling coefficients and plasma frequency ,. 2 2 2 reduction factors for beams in drift tubes, k = k' + w(jk< ) , k = 2 k, * SL r civ v = beam velocity. o 4_9 Q FUNORftENIOq T I !* I i I i i i ! I — r ;'cn"ynV voifpoj HKAM VOI.TAGK i I I 0.40 O.MS 0.50 ' 0.55 0.60 0.65 0.70 0.7S O.GO O.0S 0.90 1. INCHES Figure 7. Fundamental Component of Beam Current Along the Tube Together with Gap Voltages. TABLE V Voltage-Dependent Parameters of One-Megawatt Beams Relativistic Longitudinal Radial Beam Beam • Mass Wave Wave Voltage Current Perveance Coefficient Number Number V I K v/c k' 0 o y- r kV r ad/mm r ad/mm 50 20 1.73xl0~6 0.413 1.098 6.OSS ' 5.542 1 o o 75 lu, J 0.649 0.4S9 1.147 5.140 4.483 100 10 V, 0.316 0. 548 1.196 4.5S6 3.836 125 8 0.181 0. 595 1.245 4.224 3.395 150 6.66 0. 115 0.634 1.294 3.964 3.066 175 5.71 0.07S 0.667 1.342 3.76S 2.S07 200 5 0.056 0.695 1.391 3.616 2.600 It seemed reasonable to place an absolute limit on the beam hole size such that the TE mode could not propagate. For a 120 GHz TE cutoff, the diameter XX ! i XX is 1.46 mm, and 1.25 mm seemed a reasonable upper limit on beam hole size independent of beam coupling considerations. Figure 8 shows the radial component of the beam coupling coefficient averaged over a beam of radius r partially filling the drift tube of radius r . • Fro'm examination° o f this figure, "it is obviously desirabl"e to be able to1v/ use a small radius such as k' r = 1. However, cathode current density and r o t magnetic focusing field considerations force us to larger values such as k'. r =2 ° r o . or 3.* Table VI shows the drift tube radii corresponding to these values of k' r for voltages in Table V. Below the circuit hole'radius, r , in each square r o ° ' o of the matrix is the beam radius, r^, for a two-thirds filling factor; then the Brillouin focusing field, B^, for this beam; the area convergence, A, necessary to It is interesting to note, at this point, that for constant electric field in the beam, it is impossible to reduce the dissipation per unit area on the circuit by moving the circuit away from the beam. Because the phase velocity on the circuit is less than the velocity of light, the electric and magnetic fields on the circuit must increase as the Bessel function I (k1 r ). For most arguments of any interest, this function squared increases more rapidly than the circuit area. Also the ratio of outside diameter of the circuit to beam hole diameter decreases as r is increased. o 4-11 sin k'2/2 Where M = IvT M„ and M -- L R L • k' 2/2 r rb . I0 (k»r r) 2n rdr x / . = 2 1 r h' MR" 2 o I (k1 r ) k' r, I (k1 r ) TT r, o r o r b o r o rj-, = Beam radius r0 = Drift tube radius I - Gap length kg' = Longitudinal beam wavenumber in radians/unit length k^ = Radial beam wavenumber in radians/unit length 1.0 0.9 X 0.8 r /r = 1.0 0.7 N^O b 0.6 1.2 'J K h 0.5 I. 0 0.4 \2..C 3.0 ^ 0.3 0.2 f. 0.1 F. 1.0 2.0 3. ( kir r o Figure 8. Average Value of Radial Factor of Beam Coupling Coefficient for Uniform Density Beam Partially Filling Gap 4-12 !'.•' TABLE VI Brillouin Focusing Field and Area Convergence for Various One-Megawatt B^ams of Different Energy and Size i> Parameters for: Beam k' r = 1 k'r = 1.5 k' r = 2 „ k' r = 3 r o r o r o r o r = 0.180 mm 0.271 0.361 ,0.541 0 50 kV r, = 0.120 mm 0. ISO 0.241 0.361 b 20 A B, = 20400 gauss 13600 - 10200* 6300 b A = 14700 6520 3670 1630 r = 0. 0632 mm C o 0. 223 0.335 0.446 Z/ 0. 669 ZZ .. 75 kV 0. 149 0.223 0. 297 y 0.446 13.3 A 12090 8060,, 6040 Ay 4030 6400 2850 1600 , Z 711 r = 0. 0705 mm C u // 0.261 0.391 •> 0.521 Vy 0.782 100 kV 0. 174 0.261 0.348 y 0.521 10 A 8300 5530 4150 A 2770 3520 1560 879 V, 391 r = 0.0761 mm " // c // 0.295 0.442 0.589 Ay 0.884 125 kV 0.196 0.295 0.393 y 0.589 8 A 6180 4120 3090 A • 2060 ^ 2200 978 551 y 244 r = 0.0809 mm » "Z c 0.326 0.490 AA^.^VAAA/ A; 0.978 150 kV ,0.217 '0.326 y 0.435 0.652 6.66 A 4840 3230 A 2420 1610 1500 665 166 Y, 374 r = 0.0851 mm 0.356 0.534 A 0.713 1.069 175 kV 0.238 0.356 C y 0.475 0.713 5.71 A 3930 2620 A :1960 1310 1070 477 y 268 119 r = 0.0891 mm o c -V 0. 385 0.577 A 0.769 1.154 200 kV 0.256 0. 385' y 0.513 0.769 5 A 3270 2182 Z 1637 1090 806 ,, m 358 /> 201,° 90/^ , r - 0. 0924 mm c NOTE: Drift tube diameter exceeds 1.25 mm within shaded area. 4-13 produce the beam from a Phillips cathode having an emission density of 3 A/cm ' i f and finally, the cyclotron radius, r , in the Brillouin field for an electron % c emitted from the cathode with a transverse thermal energy of 0.1 eV and raised in temperature by the convergence of the gun according to Liouville's theorem. We intended to use convergent confined-flow focusing in any linear beam tube we might build. ^This is a form of space-charge-balanced flow in which the magnetic flux threading the cathode is almost equal to the flux in,the beam. We have found that this extremely stiff focusinguisually results in the degree of beam stability necessary in very high average-power tubes. Frequently, the extra it magnet power is more than returnod'in increased radio frequency power. For this focusing method we would require a magnet capable of Aproducin g» two to three times the Brillouin field shown in Table VI. The thermal-electron cyclotron radius ^would also be correspondingly reduced in relation to the beam diameter. Some attention was given to the possibility of building a 7500 gauss magnet, and this seemed to present no problem. An examination of Table VI indicates that acceptable values of magnetic focusing field, electron gun area convergence, rj and beam coupling exist for voltages between 150 kV and 175 kV at values, of k' __ r between 1.5 and 2.0. More specifically, the lowest focussing fields and o area convergence together with still-adequate beam coupling exist for the 1. 25 mm circuit hole diameter we,set as our maximum as a result of the TE • propagation consideration. Table VII gives various parameters for what we concluded were acceptable 1 MW, 150 kV and 175 kV beams around which 100 kW, 120 GHz traveling- r) wave tubes could be built, even allowing for efficiencies as low as 10%. Q ci a Having determined that there was no insurmountable problem in raising the beam power, we returned to the problem of circuit, loss. In the computer simu- lation of the 120 GHz tube using a 120 kV, 4 A beam, we mentioned that the computer .predicted an output of 70 kW, but also predicted a power dissipation of 2 kW in a it v cavity of the circuit near the output end. This cavity had a surface area of only 2 " " 0 2 0.04 cm and, hence, the power density in that cavity was predicted to be 50 kW/cm . o 4-14 O C o (•I TABLE VH Parameters of Selected One-Megawatt Beams Beam voltage Vo " 150 175 kv Frequency * f 120 120 GHz Beam radius .0.4167 0.4167 mm rb u Beam velocity/velocity of light v/c 0.6343 0.6672 Relativistic mass correction factor r 1. 294 1.342 Radial beam wavenumber k' 063 2. 806 S r ad/mm r Longitudinal beam wavenumber 3.962 3.767 ' rad/mm K'I Normalized beam radius k' r 1. 27 S 1.169 a racl r Beam power P io6. 10 6 W Perveance K 0. 115 0.0779' 6 / 3/2 x 10" x 10"® A/V Beam current density i 1222 1047 A/cm2 0 Brillouin focusing field Bj 2526 2238 gauss b Infinite beam plasma frequency f 3.867 3.300 GHz P v\ 4 •< & P » a a c V 7 . \ ffi ( !. lir , * I' u ,4-11 5 « The circuit used on this tube had an impedance (the square or the peak cavity voltage divided by twice the power flow on the circuit) of 400 ohms. In order to reduce the fields on the circuit, we then simulated a'tube in which the impedance 2 parameter, Z^ = V /2 P, was reduced by a factor of ten to 40 ohms. In a physical embodiment of such a circuit much of the power would probably be allowed to flow in an auxiliary waveguide. We continued to use the 120 kV 4 A beam for the sake of comparison even though we thought we might later increase the beam power. In this simulation, because of the low circuit impedance, radio frequency fields on the output circuit did not build up at all rapidly and the best efficiency that was achieved was in the order of two or three percent. ^ We then attempted, on the same beam, the simulation of a circuit in which the impedance of the circuit was tapered from 400 ohms down to 40 ohms over the 30-cavity length of the circuit. In this simulation, the efficiency only dropped from 14 percent to 10 percent. The maximum power dissipated in any cavity was 300 W , which began to approach a tolerable level, a We continued the simulations but switched from the 120 kV, 4 A beam to a 150 kV, 6.66 A beam, which we thought would give us a better chance of reacing the 100 kW level at 120 GHz. Later we found that we probably would not need this much beam power. Such a beam would be appropriate for the 0 30 GHz, 200 kW requirement, however. With adjustment of the taper, we finally reduced the highest"cavity power in a 120 GHz tube to 150 W while achieving an output power of 110 kW. Figure 9 shows the modified Applegate diagram for this tube, and Figure 10 shows the rf currents and voltages. The drive power ii was 150 W. o * This is actually the power dissipated in an area equal to that of a TMQ cavity resonant at the upper end of the passband of the circuit, and does not include losses in the transmission means used to lower Z. = V2/2 P. y k a O' o t ° 3 4-16 "0 I-1L IN 1 3 -U Cavity Numbers 4 - I- -!• i.iz ISCEX606 2. INCHES 08/11/75 Fig. 9. Phascivs Distance for Electrons in Output Section of Tapered Impedance TYVT with Output Power of 110 kW. vf^ I 0.6't O^.QII • o.sa 1 .04 1.08 1.12 LSCEX608 z. INCHES 08/11/75. Pig. 10. 1\F Beam Current and Cavity Voltages of Tapered Impcdanec TWT with 110 kW Output. We then applied an 85 percent velocity taper to this circuit (that is, the " phase velocity at the output end of the circuit was S5 percent of that at the input,, end) and the power output rose from 110 kW to 207 kW while the maximum cavity 'power roae to 375 W. Figures111 and 12 show the Applegate diagram and the voltages and currents. Again the drive power was 150 W. In the simulations, we used a cavity Q of 500 which is one-half the theoretical unloaded Q for a 120 GHz cavity of the same dimensions. In the velocity-tapered section, Q was scaled in proportion to cavity height. Values of shunt resistance divided by Q appropriate to pillbox cavities of the size used were employed, so we feel that it is likely that the parameters of^the circuit assumed in the simulation can be realized; however, this remains to be done. A conceptual sketch of the output circuit is show in Figure 13. Other configurations may be possible. -The circuit/, consists of a number of inductive slots in one side of a waveguide to'provide a phase velocity in approximate synchronism with the beam velocity. Tb&; ,width of the waveguide increases toward the output end of the circuit to provjd^'the%ecessary • Q ' 'i , tapering of the circuit impedance. The beam hole passes through the slots parallel to the waveguide wall and fairly close to the waveguide. Except for the tapering of the width to change the impedance, the circuit is very similar to a waveguide coupled vane circuit proposed by;. Feinstem and Collier9 . In addition to <> ' 0 realizing the circuit pltysically, there is a further need to optimize the phase shift per section in order to obtain the lowest loss per cavity. All of the computer ti simulation done to date arbitrarily used a TT, /2 phase shift per cavity. Circuit losses normally reach a minimum.'at a slightly greater shift per cavity than this. .We believe that this circuit could provide..the basis of a 200 kW, 30 GHz tube or a ° . . „ 100 kW, 120 GHz tube if the beam power were reduced. The input circuit would o C {l ' be a conventional slow-wave circuit with constant impedance and there would be terminations on the input and output circuits at the sever. Table Vin gives performance data, dimensions and other parameters which we believe could be achieved at 30 GHz and 120^ GHz using this approach. Table IX t gives detailed characteristics of the circuit for 120 GHz. 4-19 „ • c FEINI 30 rr<- I ; i ... i 1 1 tf1- tsO3 0 .68 01 I.08 L5CEX900 !1/03/75 Fig. 11. Phase vs Distance with Impedanec am! Velocity Tapers and 207 k\V Output 0 a.68 a.72 0.76 0.64 0.88 0.32 0.36 l.OU 1.00 LSCEX900 Z. INCHES U/03/7S Pig. 12. Beam Currents and Cavity VoILages with Impedance and Velocity Tapers and 207 k\V Output 6 Beam Hole ff 1 i \k 1 i r \ 1 i K// — r -I -•/ jS 1 1 i L^ / / r f. Section A-A Figure 13. Output Circuit with Tapered Impedance 4-22 TABLE Vin Characteristics of Possible TWT Amplifiers Frequency- 30 GHz Power output 200 kW Beam Voltage 150 kV Beam current 6.66 A Gain 30 dB Number of Cavities <•. Input Section 35 Output Section 30 Length overall 10.56 cm Beam hole diameter ® 5 mm ,Beam diameter 3.33 mm Cathode diameter 2.91 cm Cathode area convergence 76.5 Cathode current density 1 A/cm" Magnetic focusing field " 2000 gauss t< 4-23 TABLE DC r' Characteristics of 120 GHz Output Circuit with Impedance and Velocity Taper. A tube using this circuit was computed to produce 213.0 kW out for 150 W drive with a 35 cavity drive section with no impedance or velocity taper; V = 150 kV; IQ = 6.667 A; b/a = 0.6667; beam hole dia. = 0.0492 inch; fcav = 142.770 GHz; plate thickness = 0.005 inch. Gap Length z R/QW> Zk'(r=a) °l (r=a) Cav. No. Inches relative 1-6 0.0106 46.00 372.0 1.000 7 0.0106 43.19 349.3 0.939 8 0,0106 ,32.22 260.6 0.700 9 0.0106 27.58 223.0 0.600 10 0.0106 20.65 167.0 0.449 11 0.0106 18.48 149.4 0.402 12 0.0106 14.95 120.9 0.325 13 0.0106 ' 13.59 109.9 0.295 14 0.0105 11.59 92.86 0.252 15 0.0103 10.86 85.34 0.236 16 0.0102 9.93 77.3 0. 216 17 0.0100 b 65 66.0 0.188 18 0.0099 8.06 60.9 0.175 19 0.0098 7.34 54.9 • 0.160 20 0.0096 6.92 50.6 0°. 150 21 , 0.0095 6.44 46.7 0.140 22 0.0094 6.27 45.0 0.136 23 0.0092 5.92 41.,5 0.129 24 0.0091 5.85 40.6 0.127 25 0.0089 5.64 38.3 0.123 26 0.0088 5.58 37.5 0.121 27 0.0087 5.52 36.6 0..120 2S 0.0085 5.39 35.0 0. 117 u 29 6 0.0084 5.33 34.1 n. O-H '-V! 30 0.0083 5. 26 33.3 0.114 4-24 4.3 HARMONIC AMPLIFIERS '' ' ' - / An attractive alternative )to traveling-wave tubes and extended-interaction * klystrons is a frequency multiplier. The drift tube of a klystron frequency multiplier cannot provide an electromagnetic feedback path nor can backstreaming secondary electrons provide feedback. There may be a possibility that a propagating drift tube at the putput frequency could be practical if the excitation of the circuit is sufficiently uniform that the propagating modes do not lead to a significantpower loss, o Many klystron frequency multipliers have been built. Ballistic theory predicts \ fairly good performance. However, in recent years there have been substantial "improvements in multicavit•J y klystron efficiency by proper choice of cavity spacing and tuning. The improved bunching in these klystrons implies increased harmonic beam current components. To verify this, we decided to run some klystron cases on our large-signal program. We chose a 1 mm diameter, 120 kV, 4A beam.Q The drift .tube was 1. 2 mm in diameter. A 30 GHz bunching frequency was used. Best performance was obtained with the,cavitieo s spaced and tuned as shown in Table X. TABLE X Cavity Tuning for Harmonic Amplifier Distance from Resonant Cavity Cavity 1 Frequency Number (cm) (GHz) 1 0 30 2 10.7 30.27 3 18 . 3 30 . 14 Figure 14 shows the modified Applegate diagram for the tube and Figure 15 shows the development of the fundamental, second harmonic arid fourth harmonic components of current. The fourth harmonic current component reached a maximum This concept was suggested by Dr. Marvin Chodorow, consultant to Varian. The authors gratefully recognize this contribution. I' II o r fied Applegate Diagram Showing Performance on a Tube with Cavities Spaced and Tuned town in Table VIII. „, % u- ftli- 4-27 2.S 3 .0 3.5 4.0 4.3 5.0 s.s 6.0 6.5 7.0 Z. INCHES 5. Development of the Fundamental, Second Harmonic and Fourth Harmonic Components of Current BLAN<7 K PAGE of 1. 52 times the clc beam current at a distance 2.15 cm beyond the third cavity- gap with a drive power of 250 W applied to'the input cavity. Dissipation in Cavity reached a level of 5 kW, but since this cavity has about 16 times thenarea of a 120 GHz cavity, the power density may be tolerable. The 120 GHz current in this multiplier is roughly equal to the, 120 GHz current in the previously-described traveling-wave tube, and so the idea of a frequency multiplier with its inherently-simpler stability problems seems attractive. The problem of reducing output circuit dissipation remains, however; and for the reasons outlined in the introduction to this section, we decided not to pursue this approach further. , , oO s. % t> 4-31 5. PERIODIC BEAM DEVICES 5.1 INTRODUCTION /J A search was conducted for literature relating to periodic beam inter- actions that might be suitable for high-power millimeter-wave amplification. A particularly thorough discussion of possible interaction mechanisms was found in the paper by Gaponov et al10 . This paper discusses a large number of micro- wave devices from the point of view of excited classical oscillators. It deals with CRMs (cyclotron resonance masers or gyrotrons) operating both at the fundamental cyclotron frequency and at harmonics. It discusses CRMs employing helical beams in uniform magnetic fields and those based on trochoidal beams in crossed electric and magnetic fields. It also considers'the Ubitr-on and the Strophotron as possible millimeter wave amplifiers, and it includes an extensive list of references. Another reference that has a very complete discussion of approaches for millimeter-wave generation is by Kulke and Veronda11. This article discusses results obtained with both linear-beam and periodic-beam devices. It contains a bibliography of 77 references on the subject of millimeter-wave electron-beam devices. IT ' ' The literature search led to conclusions in agreement with Gaponov et al.-' o that the cyclotron resonance devices offer potential advantages over other fast- wave devices for high-power„ , high-frequency amplifiers. Accordingly, the emphasis in this study was on these devices, and an entire section of this report is devoted to them. Other periodic-beam devices which were studied to a lesser extent will be discussed in this section. These include the Ubitronl12'13 and the periodic-beam TM interaction of Dyott14 >15. 0 0 • . V.\ ' 5.2 UBITRON The Ubitron12»13 in its most interesting form uses an undulating, cylindrical, hollow, electron beam interacting with TE^ cylindrical waveguide fields. The beam undulation has the form of a corkscrew pattern periodically alternating clockwise BLANK PAGE and counterclockwise. A"millimeter-wave pulsed version of the Ubitron operated at 54 GHz with a power output of 150 kw peak and 250 W average. Beam parameters were 60 kV and 35 A. The conversion efficiency was 6%. With the Ubitron, the strength of the interaction and, hence, the gain and efficiency depend on achieving a significant amount of undulation of the beam. The energy given to the wave by the beam comes from the azimuthal velocity of the electrons. The undulation is obtained by a periodic magnetic structure which can take the form of a set of iron rings immersed in a solenoidal field as shown in Figure 16. The rings must be spaced with a period of a free space wavelength or less. One of the problems of this system is that the variation in magnetic field decreases rapidly as one moves radially inward from the inner radius of the rings toward the radial position of the beam. For a beam of finite thickness ^the inner electrons are undulating less than the outer ones and therefore are not effective in the interaction. "/ Another problem is that at the radius just inside the rings where beam undulation is best, the E , field of a smooth guide is zero. Enderby and Phillips13 P solved this problem by cutting axial grooves in the inner wall of the waveguide. These grooves move the radius of the maximum E^ field out closer to the wall containing the rings. • * v y ' "ii The magnetic field required for undulation can be estimated by noting that the B^ component must convert a significant amount of the beam axial velocity into azimuthal velocity in a space comparable to a half free space wavelength. This requires a radial component of magnetic field in the region of the beam of about 1/4 of the cyclotron field. ll c? One of the advantages of the Ubitron over the fundamental CRM devices is that it is not necessary to apply the cyclotron resonance magnetic field. However from the arguments above it can be seen that for higher frequencies, stronger fields and shorter magnetic periods must be used. The"54 GHz Ubitron. employed Magnetron Input Coupler Injection Rectangular to Solenoid Gun Circular Waveguide Output Window Laminated Steel Solenoid and Copper Interaction Circuit Figure 16. Schematic Drawing of V-band Ubitron Amplifier 0 a solenoidal field of 10,000 gauss compared to a cyclotron resonance field of 21,900 gauss/T) It is probable that this ratio would have to be maintained in frequency scaling. The problems in achieving a uniform undulation of the beam over its cross section would appear to be more difficult in the Ubitron than in the CRM devices o where a natural orbiting of the electrons is needed. It appears likely that the low efficiencies obtained with the Ubitron may be caused by this problem. Although the Ubitron is not ruled out for the desired application, it is considered to be less advantageous with respect to the cyclotron resonance devices. <> 5. 3 PERIODIC BEAM TM INTERACTION An interaction involving a rippling beam and a TM^ cylindrical waveguide field has been reported by Dyott and Davies14»15 . Apparently the field of importance in this case is the E field and a synchronous, component is developed by causing z the electrons to orbit in and out of regions of high E fields. The device is explained z on the basis of axial bunching of the beam and exchange of axial kinetic energy. However, it is pointed out that other fields may be important such as E and H , rf r p fields. o ... n " This device requires a;dc magnetic field only slightly less than the .cyclotron field, hence it is not advantageous in that resoect over the CRM devices. The reported device was operated as an oscillator. Typical parameter values were a power output of 36 kW at 40 GHz with beam voltage of 80 kV and current of 6 A. The resulting efficiency was 7%. It was pointed out that the center.,,. of the beam is ineffective in this device because the gradient of E is small near z the axis. The efficiency might have been improved if a hollow beam had been used. 0 A possible advantage of this device is that the energy for microwave output comes from the axial kinetic energy of the beam which is the most natural form for e f the beam energy. There is no need to convert most of the energy to transverse J- 5-4 energy as is required in the CRM devices. Velocity spread should be much less of a problem in this device*. ! The power-handling capability of the TM^ guide may be limited by either loss power density or by electric field breakdown. The loss power density can be determined from Figure 1. We will consider the case where f c /f = 0.8, then the ideal copper losses at a wavelength of 2.5 mm are 3 dB per meter. With a 2 100 kW power flow, this translates to a loss density of 665 W/cm , which is an allowable value. a j The relation between power flow and axial electric field is given by : where f is the cut-off frequency, a is the guide radius, is the electric field amplitude on the axis, J is the first order Bessel function and k = 2 - f /c. If- ^ r- ® Putting in the values P = 10° W, and f /f = 0.8, we get E = 1.02 x 10' v/m, c zo which is an allowable value. The radial electric field at the wall is related to the axial field on axis by1: ^ E .= E J ra" zo 7 In this case, we get E = 0.40 x 10 V/m, wiiich is also acceptable. rao e> One disadvantage of the periodic beam TM device is that the full value of the electric field is not effective in the interaction. As Dyott has pointed out15, if the transverse energy of the beam is 80 kV, the effective E is 0.3 times the 7 n Z peak value., Assuming a peak field of 10 V/m, the effective voltage developed per wavelength at^>. 5 mm would be 7.5 kV. Then, to develop SO kV of circuit voltage would require about 11 free space wavelengths. 5-5 a Another disadvantage when compared to the gyrotron devices is the smaller beam cross section available. The TM , cutoff at k a = 2.405 compared 0rt1 c to the TE cylindrical cutoff at k a = 3.83 results in 60% more beam area for 1) i. C the TEq^ device. Conversely, the periodic beam TM device has more beam area than a conventional linear-beam amplifier. /v 6. CYCLOTRON RESONANCE DEVICES 6.1 GENERAL DESCRIPTION OF DEVICE ALTERNATIVES < Amplifiers and oscillators based on the principles of cyclotron resonance have been the subject of investigations spread over a considerable time period. We will not present a complete listing of references on this subject, but will discuss several that are representative. P A cyclotron resonance devicelin which the electron motion was determined by crossed dc electric and magnetic fields was described by Swift-Hook and Reddish 16 . An oscillator based on cyclotron resonance with dc magnetic field was built by Chow and Pantell17. A discussion of cyclotron devices from the point of view of maser action was presented by Hirshfield and Wachtel18. An oscillator which produced watt power levels at millimeter wavelengths was reported by Bott19 Similar results with related devices were presented by Beasley20 . The Hirshfield, Bott, and Beasley devices all used corkscrew magnetic fields to produce transverse electron velocity followed by magnetic abiabatic compression of the beam. The millimeter C devices used superconducting solenoids. Work on X-band cyclotron wave devices was done by Schriever and Johnson21 at the University of Utah. This work resulted in a traveling wave device, which could operate either as an amplifier or an oscillator. Measured gain as high as 20 dB was achieved and power output as an oscillator was 720 W at X-band. Another X-band oscillator using a short cavity resonator was reported by Kulke and Wilmarth22 . Recent work by Granatstein23' 24 has dealt with cyclotron resonance or synchrotron effects with very high peak power, short pulse, electron beams. Power o levels of 50 Mw peak at 8 GHz were obtained, for example, with a beam voltage -of about 1 MV and useful beam current of about 1500 A, for an efficiency of 3%. Pulse lengths were of the order of 50 nsec and pulsing was nonrepetitive. Peak • \ 6-1 1 c? power of 'the order of 2 MW in the 60 to 90 GHz range has also been obtained with similar beams where the output frequency is a high multiple of the cyclotron frequency. Sprangle and Granatstein25 have also proposed another technique of obtain- ing submillimeter wave power by scattering from an intense relativistic beam. Incident microwave radiation at a lower frequency is scattered and doppler-shifted up in frequency by scattering off the high energy beam. Very high voltages and high currents are needed to achieve appreciable frequency shift and reasonable efficiency. Experiments to pursue this approach are reportedly planned. 0 The record for high cw power (near 100 GHz) reported in the literature at this time appears to be 12 kW cw at 2. 78 mm wavelength. This was reported by Zaytsev et al . This power was obtained with a cyclotron resonance oscillator (gyrotron) using the fundamental cyclotron frequency interaction. Because this c is a significantly higher power level than achieved by other approaches, we will discuss this work in much more detail. The configuration of the device is indicated in Figure 17. The interaction uses a hollow cylindrical electron beam where indi- vidual electrons have helical motion. The microwave structure is a cylindrical TE resonator with a fairly large length-to-diameter ratio. (Actual dimensions UZ1 are not given in the reference.) # Beam parameters are reported to be 27 kV and 1.4 A, and the electron o axial velocity was given as 2/3 of the rotational velocity. For 27 kV, the total electron velocity is 0. 315 c ( where c is the velocity of light) and the rotational velocity v is 0. 262 c. The equilibrium electron orbit radius can be obtained from the equation: v, 7 m r ^-Hr-2- • (4) e eB v ' where y is the relativistic mass factor. % ^ 6-2 u <• 5 « " 1 - Solenoid 2 - Cryostat 3 - Injector 4 - Resonator 5 - Collector 6 - High Frequency Window © ( 6-3 For the beam parameters in the gyrotron with an applied magnetic field of 40. 5 kOe, we get an orbit radius of 0.116 mm. The cavity radius for a long TEq21 cavity resonant at 2. 78 mm wavelength is 3.1 mm. The actual beam cross section used in the experiments was not given. It is assumed that the beam was a hollow cylindrical beam. Based oil this information, the cross section of the C" TE interaction mus t be approximately as shown in Figure % 18. The figure shows i. the caivity and electron orbit size to scale, but the beam cross section is only esti- „ ' y s mated. o " !> - In the gyrotron interaction, it is^clear that energy exchange between the' ft • electrons and the microwave fields must occur by virtue of an E' field in the local coordinate system for the orbiting electron. For the TE^9 interaction, the local E ©, field is also the E 0J field in the cavity coordinat" e system. For maximum inter- action, all electrons should be located in a region where the cavity E has a maximum value. The hollow beam cross section shown in Figure IS satisfies this condition. Note that the entire cross section of the bekm is effective because the cavity fields have'"no variation in angle. This effective use of the entire beam cross section is one reasonn for the high efficiency of the fundamental gyrotro^ n ii device. The measured efficiency in this case was 31% for conversion of beam power to microwave output power. o o Operation of a gyrotron oscillator using was essentially the same as that for thOe 'Ifundamenta l gyrotron shown in Figure 17 6-4 Scale 1 mm I TE, Figure 18. Cross Section for 021 Gyrotron 6-5 There is some concern as to whether a linearly-polarized TE^^ field can effectively interact with the entire beam cross section. To analyze the interaction, we must consider the cross section which is shown in Figure 19. Figure 19A shows the variations in,E , and E with radius. Figure 19B shows schematically the dis- monic interaction, the field polarity.re verses 'and returns too its original polarity while the electron travels through half its orbit. In a region where the gradient of O the field is zero, there will be negligible interaction because the average force ,. applied by the field will be zero to first order. " s ' Reference to Figure 1"9 shows that there are four .angular positions where the E , gradient is maximum and four where it is zero with a sinusoidal variation ^ ! with angle. The E gradient has similarcbehavior with angle except, that'it is • shifted with respect to E^ by 45°. T/uere exists a range of radius where the gradiento s of E^Jo and E r with respect to radius are about equal. In that case,o ' when we suni the two "Vector components in the local coordinate system for each electron,', » we obtain a constant value independent ofi angle. -If the E^ and E^ gradients'were " not equal the vector sum would be a constant plus a sinusoidal variation. It is con- eluded that the linearly-polarized fields can be reasonabl-y effective ovenrf the entire beam cross section in the case of the TE„„., mode. s 231 0 ° fj • • It is possible, also, that a circularly-polarized TE mode was set up in 0 0 231 ° the device. The circularly-polarized fields can exist whenever there is perfect & •it ° v 6-6 * f 0 '•' o Beam Cross Section - Electron Orbit 0 - - „ < ; Beam Cross Section Electric "Field Lines 0 Figure 19. / Cross Section for TE9g Gyrotron. 6-7 11 T ftsri' f • / ' ' 11 '•}) cylindrical symmetry in the cavity and in the output coupling system,'.i which was :: apparently the case in the Zaytsev •experiments '•;) . ' For the circularly-polarize'1 d ca&,e, the field pattern in Figure 19 simply rotates one-half revolution intone rf period. v > (• I "/V '' - ' Then, electrons at any angular position can be exposed to exactly the same value^ o • " of field. The rotational slipping of the fields past any given electron is of'no J particular consequence except to determine the relative phasing of the rf electron current. In this case, all angular^ positions in the beam are fully utilized. For the TE mode, all angular positions of the beam can be used effec- <>tively since the fields have no angular variations. The TE „ ' mode has ho E but J ° 031 r has a radial variation of E , similar to the TE„„ . For both modes, the optimum (p • 23H1 radial position for the beam for second harmonic operation is .nearly the same. The results of Zaytsev indicate higher efficiency (15%) for the TE opera- <, ' '' ' Cl*J -L tion as compared to the TE operation (9. 5%). He also gives calculated values U o J_ ° of 20% and 15%, respectively, for the two modes, based on a reference which we have not been able to obtain. From these results, it is concluded that utilization of the full beam area is not an important difference between these two'.cases. Pre- sumably, a parameter such as fiel•a d strength is the important factor, which may vary because of cavity losses or load coupling. h G 0 u <' b Another very impressive gyrotron oscillator employing second harmonic operation has been reported by Kisei' et al27. The device was similar to thec-one built by Zaytsev, indicated in Figure 17, except that a room-temperature solenoid was used. A power output of 10 kW cw at 8. 9 mm,wavelength was obtained with an efficiency of 40%, With pulsed operation,' the output was 30 kW at 43% efficiency. The excellent efficiency of this device was attributed to the use of special shaping techniques for the microwave field in the oscillator cavity. This field shaping will be discussed further in a later section.on,gyrotron theory. The essen- ce ' tial feature of the field shaping apparently is to apply lower electric field in' the beginning of the cavity where electron bunching is taking place and higher field 6-8 later when energy is being removed from the bunched beam. The field shaping is accomplished by varying the cavity diameter. The cavity shaping is indicated in Figure 20, which shows the cross section of the experimental oscillator. Another approach for cyclotron resonance devices is to employ a beam in which each electron orbits about a nearly common axis. Then one can employ a circularly-polarized cavity mode, for which all electrons see essentially the same average component of E . The disadvantage of this approach is that for low beam This approach of orbiting electrons around the cavity axis also allows ir relatively-strong interactions to occur at high harmonics of the cyclotron frequency. Experimental devices using harmonics up to the 13thharmonic were demonstrated by Jory28. These devices were based on interaction with TE cylindrical fields. The principal field of interest is given by1 : ^ nil COS n E = A 7/i/n e ~ J' (k r) cos (p Z) / _f „ (5) *-'f V/-^2 * < ' „(6) r J' (k r) indicates the derivative of J with respect to the argument, k r. The n c n c variation of E, with radius is indicated in Figure 21 for the cases of n = 5, 10, 'and CD e> 20. It is observed that, for high values of n, the E fields are concentrated in a relatively-narrow region near the cylindrical wall of the cavity which will be at the radius where E^ goes to zero. 3 1. Gun Structure 2. Emitting Area 3. 1st Anode o 4. 2nd Anode 5. Cavity 6. Output Coupling Iris 7. Collector S. Vacuum Window 6 9. Output Waveguide 10. 12. Water Cooling 11, 13. Solenoids 14. Ceramic Insulators Figure 20. Drawing of Experimental Gyrotron and the Axial Distribution of the Magnetic Field (alter Kisel1) 0.16 0.14 • 0. 12 n = 5 j U. lu r. \ 0.08 \ < 10 i \ 0.06 20 / 0.04 f D /I 0.02 0.2 0.4 0.6 -.8 1.0 1.2 k r f v c JL n f ,. c Figure 21. Radial Variation of E , for TE , , Cylindrical ^ © 6-11 To maintain synchronism with a circularly-polarized component of the cavity field, an electron must rotate about the cavity axis with angular velocity d(p/dt ="oj/n. Then the tangential velocity of the electron must be v = rw/n. If these relations hold, the abscissa in Figure 21 can be identified as equal to f v /fc, or essentially. v,/c. Then it is clear from the figure that large values of v are needed to put electrons in a high field region, particularly as the harmonic index, n, gets large. The potential advantages of this interaction with a large value of n are that the magnetic field for synchronism at a given output frequency is reduced by l/n and the allowable beam cross section increases in proportion to n. Disadvantages are that the beam cannot be placed at the position of maximum E . © and high beam voltages are required to avoid significant decreases in E amplitude. 6.2 ANALYSIS OF CAVITY LOSSES IN CYCLOTRON RESONANCE DEVICES As was pointed out in the prior discussion of linear-beam-amplifiers, r the magnitude of the electric field available to interact with the beam is limited by tolerable circuit dissipation. In that case it was not possible to reduce the it magnetic fields at the wall-by moving the wall away from the beam because of the constraint that the phase velocity of the wave had to be less than the velocity of light. For cyclotron-resonance devices, cylindrical cavities are used, the cavity fields are standing waves and, for some modes, it is possible to reduce the magnetic fields at the walls by moving the walls away from the interaction ^ • 0 region. For example, the 2 circular electric modes have magnetic field, at the cylindrical wall which may be made as small as one likes by increasing the diameter of the cavity while simultaneously going to higher mode numbers, m. Also, if one picks a cavity resonant frequency close to the TE. waveguide.mode cutoff, the cavity end walls will be farther apart and the magneti0,cm fields at thes(j e ., ' walls can be made lower than^if the cavity resonance is far above cutoff. There are other modes available for cyclotron-resonance interaction a"' which do not have the characteristic described above, however. An example is the TE^ ^ ^ family which is useful in devices operating at harmonics of the . cyclotron frequency as described by Jory28 . In this mode family, for large cavity diameters and large values of n, the electric field tends to be one-quarter wavelength away from the cylindrical wall. On the other hand, as n is made ti small the maximum value of electric field increases in relation to the magnetic field at the wall, and the distance between the wall and the e(lectric field maximum increases. In general, we believe it is correct to state that for TEn m Q » if n is increased the ratio of maximum electric field to magnetic field at the wall will r. O be decreased; if m is increased, the ratio will be increased; and the value of 1 has no effect. However the fields at the end plates may be made low by choosing the proportions of the cavity so the cavity resonance is just above the cutoff of the TE waveguide mode, n, m u In order to arrive at a quantitative evaluation of effects described above we have devised a figure of merit for cylindrical cavities supporting transverse electric modes. The figure of merit is a normalized ratio of the square of the 2 . maximum electric field, E , to the power lost per unit.area, P /A, or (7) 2 in which o is the skin depth, A is the wavelength, Q = coU/P , H/Q = ^ TL' 2wU V = AE , U is the stored energy, and oj = 2/rcA. For TE„ . mode, from standard sources on waveguides29,30.with a 0,m,l little effort, one can show that 9 • e jroo ^OoT , (J. L R/Q = v o ! \_L, (8) 2 ? 1 3/2 o ^Onr ' 6-13 3/2 + l 7r t> L (9) Q 2 2'M + * £ tr r Om\ a (t) + (10) 4 (t) famY a' _1_ j;>oi> 2 + 7T (f) (id rfWr J 1 with respect to kr, and p"n for example, is the m^ value of kr which makes J^ (kr) = 0. Equation 11 is plotted in Figure 22 as a function of D/L for several TE , modes. r0t , m, 1 if For TE , „ modes, n, 1,1 2 2 16 (p» ) / — J» vt(p'» ) (a W. e nrn R/Q = o (12) 3/2 tr2 (p'2 -n2) J 2 (p» ) nm n nm 3/2 1 n f P \2 1- nm + (t) nm.7 (13) 2a a n + 1 nm/ ss> 6-14 8 10,000• TE08, „ o - TE '041 - « j rE 021 1000 . lS. (M o W 9 f \ \) 11 ""SJ « « o 0 a 100 0.1 „ 1.0 10.0 d/l . - .. ' . Figure 22: Figure of Merit'for TE Modes O.m.l (R/Q)(Q°/X)(A/X2) = vsD/L c£t 6-15 'fP 'nm AV A A - 4 iii (14) a L Jn' (Pri" m) (15) J ~(p' ) n nm Equation 15 is plotted in Figure 23 as a function of D/L for several TE; modes. n.1,1 6.3 GYROTRON GUN DESIGN A fairly-limited amount of information has been found in the literature concerning specific design numbers for the electron guns used in gyrotrons. Zaytsev et al26 describe the gun used in their experiments as an adiabatic gun and show a sketch indicating a magnetron injection gun configuration. The sketch shows a control anode, which presumably runs at a potential in the range of 1 to 10 kV above cathode. Curves are shown of power output and efficiency as a function of control-anode voltage normalized to a critical value with beam current as a parameter for the various curves. (i e • r; In interpreting these carves, one^rnight question how beam current was kept constant as anocle voltage was varied. A study of the other Russian references dealing with beams suggests that these gunos are operated with the 11 • cathode temperature limited. In that case, the main effect of changing anode voltage is to change the distribution of energy between transverse and axial velocity. Presumably, the heater power may be adjusted slightly to keep beam current constant, but magnetic field in the gun region is presumably not changed. The critical value of anode voltage is defined by Avdoshin et al31 as the value for which the electron orbit diameter in the gun region becomes equal 6-16 17 1/ 1000 'Ol r< 100 ft 10.0 Figure 23. Figure of Merit for TE Modes Qs n, 1,1 (H/Q)(Qf)(AA2) ^yi^ I vs D/L J 6-17 to the cathode-to-anode spacing. At this point, the anode current will begin to rise sharply as anode voltage is increased. The same reference specifies the operating point (which presumably is very close to the critical point) as that where the anode current is 10 percent of the cathode current. Returning to the Zaytsev curves where anode voltage is varied, it is concluded that the curves demonstrate the sensitivity of power output and efficiency to transverse velocity of the-electrons. A 10 percent change in anode voltage, which produces a 10 percent change in transverse velocity, causes a significant change in output power. Si The name adiabatic gun clearly has reference to the magnetic compression emploj'ed between the cathode and the interaction region. This compression is advantageous with respect to minimizing solenoid length as well as increasing o a b cathode area. The amount of .magnetic compression or the rate of compression are not described. It is probable (but not entirely clear) that considerable 'clc u () axial acceleration takes place simultaneously with compression. If thai; is the case, then the distribution of energy between axial and transverse velocity will be modified accordingly. « ; i' u Gol'denberg and Petelin discuss adiabatic compression with a magnetron injection gun in more detail. They give the velocity components in the interaction region of the gyrotron as: e E 3/2. k v±Q = ^ c — (16) 0 (17) where a = H /H is the compression ratio. H /H, refer to magnetic fields. The O K O subscript o indicates quantities in the interaction region; and k indicates values at the cathode. E is the transverse field applied at the cathode by the control anode voltage. U is the total beam voltage. These expressions neglect space charge in the gun, but do account for axial acceleration during compression. o 6-18 The same reference discusses the effects of thermal velocities and space charge in causing a distribution in the transverse velocities of the electrons. It.is pointed out that a scatter in transverse velocities of 20 percent o will generally cause substantial reduction in gyrotron efficiency. The following expression is given to determine the spread in transverse velocity at the cathode caused by thermal velocities and cathode surface roughness: o (18) where is the angle between the cathode surface and the magnetic field; rC 2rj. is the maximum distance traveled by the electrons measured from the ( given by Gol'denberg and Petelin in the form: c Av (19) I j-k P where I is the operating cathode current and I is the cathode current that would be obtained if the gun were'operating space-charge limited. To keep "the velocity spread small, it is concluded in the reference that the current ratio I/I must be small. This presumably implies temperature-limited lt P 'operation of the cathode, which^is a disadvantage in.terms of required-cathode area. , _ The space-charge-limited tcurrent for the magnetron injection gun was evaluated in the same references using two approximate cases: where the a 0 6-19 cathode length in the axial'direction is short, ana where it is long compared to the distance Fp^ short cathode case, the limiting current is: . "F> v. • i^kRE.2 f -A^fc' k k (20) >ifi.. P '1f#|3H '-'Si1!' f & 0 and-for''the long ca:ili<3'ri% 0 v. .. k k k k (21) where is the axial length of the cathode, R. is the average radius of the u cathode surface, and the other parameters are the same as defined above. Gol'denber.• Anotheg anrd possiblPeteline alimitatios n onj^eam current in gyrotronsjis given by ii % 4U 7 (e/2m)U • o N x ' Ho I (22) £n N(R /RJ max p o This limitation arises by virtue of space-charge depression of potential in the interaction regiont is the drop in beam potential due to space "charge which musf be a small fraction of the beam voltage, U , in order to have < > o ° ° 0 " a , gooa efficiency. U , is the axial equivalent voltage, R'- is average beam radius, o ll o o and R is the circuit radius. This expression assumes a thin hollow beam,. - P o & u © The .conclusions drawn are .that for millimeter-wave gyrotrons, space- ' t. a . charge effects in the gun are an important limitation, but space charge in the ... Q - interaction region is not. Limiting current in the gun will increase if cathode ° e - field E is increased. Even though the increase in E ^requires a correspondingly 0 „ larger H. to keep transverse energy constant and therefore a reduction in Cv k o ^ -v.? „ cathode radius R, , the net result is still an increased I . The limitations,in I k j- • , p P for either short or long magnetron'guns are nearly equivalent. 0 c 6-20 A publication, by Tsimring presents the detailed derivations and back- ground for the expressibns given by Gol'denberg and Petelin with regard to 0 o " - ... u i ^ thermal velocities and cathode roughness. It also discusses secondary emission and surface activity variations which increase the velocity spread in real f7j t • >'' ' , cathodes. Representative velocity spread values are given as 14 percent for " '' ' v 1 5 typical oxide cathode magnetron,guns,". 10 percent as a,contribution from secondary,emission',' J ' Sipercent as^resultin1 g from a surface roughness of 2j.tm. •V-'-i' • .- , ' M iy'"Mjv « . .'Z' >' ' I ••.•'A 't of, transversy The publicatioe velocityn spreadby Avdoshis in magnetron et al ngive guns sexperimenta with adiabatil cmeasurement compressions using a number of different cathodes. Measured velocity spread for yttrium 0 oxide cathodes haying 20 pim surface roughness was 17 percent. A spread of • & 11 percent'was predicted from roughness and 4 percent due to the operating temperature of 160,0° G" A smoother, pressed, yttrium, oxide cathode had a measured spread of 9 percent with'5.^m;;surfa"ce,roughness. A\sintered ' • 1 barium-oxide cathode had a measured spread \pi 17 percent, while pressed a oxide or tungstate cathodes had spreads of 9 percent.' <>" • 11 " " Tlie same reference presents measurements of the ^spread"3 i'h transverse, velocity caused by space charge in the magnetron injection gun.. It was,.found that the added spread due to space charge is about 10 percent when the.cathode0 current is 10 percent of the Langmuir current. ,The Langmuir current is defined as the current'that would flow to the anode under space-charge-limited conditions with no axial magnetic field. > '">, , , „ .V a •• o r- ,i '' - " % \ r, .Another approach to obtaining a beam with high transverse, energy' was o - - • o j, , i used„by Hirshfield and Wachtel18 and also-by Bott19. This approach uses a , 1 " - 'a ° ' conventional axial gun followed by a corkscrew magnetic field \vhich converts , o <• ' 1 o -r axial velocity to transverse velocity bv means of periodic transverse magnetic fields. The corkscrew region is followed by magnetic compression which further increases transverse velocity. A problem with this approach is the radial variations of field strength that are inherent in the corkscrew field. This results in a radial spread in transverse velocity which appears to be unavoidable. Tlie extensive pursuit of the magnetron gun by the Russians, combined with their success at millimeter-wave power generation, leads to the conclusion that the magnetron gun is a preferable approach. „ Still another approach to form a beam with high transverse energy is a conventional axial hollow-beam gun followed by a magnetic field reversal. This approach was used by Beck et al34. This results in a beam for which all T electronsnorbit about the beam axis. It is not a suitable beam for the EQml mode fields used,in the gyrotrons, but does have application for cyclotron harmonic interactions. This is discussed further in the next section. 6.4 AXIS ENCIRCLING BEAM , c Because the TE . device investigated bv Jory23 offers the attractive o n, 1,1 possibility of operating at a harmonic of the cyclotron frequency and, consequently, with less magnetic field, we decided to investigate the possibilities of generating the required axis-encircling beams. We pursued this work until we discovered that cavity losses for the TE n, 1,1 mode, s representeu d a substantial barrier to the use of this device as a very high power source (see 6.2). The work is reported on below. ,, o * Two type0s of axis-encircling hollow beams which might be useful in a TE „ , device can be conceived ofl They are shown in cross section in n,l,l o Figures 24a and b. Figure 24a shows a laminar beam in which electrons toward the outside of the beam have progressively larger orbitCP s with greater transverse ° u energy. Ideally, the axial energy would be equal for all electrons if the beam is to be used in a multicavity amplifier. For a single-cavity oscillator this is o not so important. Electron flow of this kind occurs in Brillouin-focused solid beams because, for the parabolic space charge potential depression, the total energy of the inside electrons is reduced by exactly the necessary reduction in rotational energy. When the center of the beam is removed the space charge a 6-22 o o 24 a 24 b If Figure 24. Possible Electron Orbits in Hollow Beams "Useful in TE , „ Devices. n, 1,1 11 <, conditions are altered and, it would seem, that laminar flow is only possible for infinitely-thin beams (in which case laminar may be a misnomer) or for ') i thick beams in which there is a spread in axial velocity. " Figure 24b shows a beam in which the transverse energies and axial energies arc essentially equal, but the orbits are not concentric. It may be possible to form such a beam from a gun or an electron lens which produces, trajectory crossovers. y A physically-realizable beam will probably have a mixture of the characteristics shown in the figures. One straightforward way to produce an axis-encircling-beam is to put a hollow beam through a magnetic field,.reversal. If this method is used, there is a limitation on the( size of the cathode when the flux inside the cathode a radius and the final orbit size in the final magnetic field have been specified. This limitation may be derived from Busch's theorem. The exact nature of the magnetic field reversal is also important. Both these subjects are discussed below. Cathode Size Limitations ° 0 0 u An electron, emitted from a cathode at a radius r from the axis of • o o ^ an axially-symmetric magnetic field, will trace a trajectory which usually o will be helical if the strength of the magnetic field changes along the path of the electron. (Exceptions.include the axial electron and some lens systems in which the flux inside the trajectory is the same before and after the lens.) If this helical path is rotated about the axis of the fielcl, the trajectory will generate a surface which containb s the trajectories of all electrons emitted from an annular cathqde of radius r . Such a surface is shown in Figure 25, It is typical of the trajectories that would occur if the magnetic field direction is reversed near the fir'st scallop. It can be shown that, in general, annular cathodes of different radii can produce, for beams in the same final magnetic field, trajectory surfaces which will intersect. c 6-24 Also, an annular cathode of fixed dimensions can produce beams of many different transverse energies in the same final magnetic field, and each will be characterized by a different trajectory surface. The transverse energy for a given cathode radius and magnetic field wj.ll be determined by the electrostatic and electromagnetic design of the electron gun. Busch's theorem may be used to investigate the characteristics of the trajectory surfaces. One approximate form of Busch's theorem is given by Pierce". (2.3) where i] is the electron to mass ratio, B^ is the axial magnetic field in the beam, and B is the axial magnetic fie'ld at the cathode, r is the zo o trajectory radius at the cathode and r is trajectoi-y radius in the beam, both in the field frame of reference. For an electron orbiting in the beam, the value of 0 in the field frame of reference is obvious from Figure 26, which is a projection of the electron trajectory in a plane normal Jo the axis of the magnetic field. As a result V Equation 23 can be rewritten 2 ( 'Y — cos6=-J B - B -7 O (24) r 2 \ z zo 2 / in which is the tangential electron velocity. The maximum and minimum values of r are given setting cos d\ = i'. If we also let B = -kB , then zo z "equation 24 becomes %V ^ 4 » „ (25) a v.* 6-26 n h Q I ..(I 0 Figure 26. Coordinate System for Electron off Axis. i> 6-£7 2 2 r - 2r r + kr =0 (26) c o (27) K rc in which r^ is the cyclotron radius. Equation 27 is the equation of a circle as shown in Figure 27. _ field at cathode B ~ final field z Cyclotron radius „, Radius of electron at cathode Figure 27. Locus of Allowed Solutions with Magnetic Reversal ^ J ,-n Q The interpretation of tins plot is that for a value of nk — =1, the c • trajectory surface does not ripple and has a radius equal to the cyclotron 0 ro radius for a transverse velocity v ,and a field B . For values of 'v/Tc — < i , y t ° z . © c a rippling trajectory surface will be produced with maximum and minimum radii larger and smaller than the cyclotron radius as given by the intersections 0 c ' - ' rQ of a vertical line with the circle of Figure 27. For nk r- > 1 ,cno solutions •Q 6-28 exist which give a cyclotron radiustas small as v /'77B ; only beams with a « t z i—i 0 cyclotron radius large enough to make n k— ^ 1 exist., A c ' Nature of the Field Reversal O <1 J., (' Figure 25 is typical of the kind of beam one^ets when one puts a smooth 0 hollow beam through an abrupt magnetic field reversal. In fact, it may be less rippled than one deserves if no attention is paid to the nature of the reversal. 36 * This problem has been studied by Burke in connection with periodically- focused electron beams. He found that if an appropriately-sized dip in the magnetic field were placed one-half a cyclotron wavelength in front of the reversal, a smooth beam would result. He also investigates other compensation schemes, such as two smaller dips one-hali'wavelength on either side of the reversal. For our problem the single dip seemed appropriate. A number of .field reversals of various shapes were tried using a relativistically-correct computer program. The initial hollow beam had an Li inside radius of 2.5 mm and an outside radius of 3.0, mm. The beam energy was 200 kV and the current was 2 A. The charge density was distributed « uniformly. • This beam would be suitable for a device operating in the TE J.Un , J., J. mode1; Figure 28 shows what happens to the beam in a simple reversal. While tV ... . there is interception of the beam on the polepiece and mirroring of. electrons, outside a 2.75 mm radius, this is to be expected from the analysis in the prior section. What is important is the scalloping. Figure 29 represents about the • o , O Q W >1 best we did using a compensated reversal. In this case, the scalloping was substantially reduced by introducing a dip in the magnetic field with a "floating polepiece" of 4 mrn^inside radius and 4.8 mm length. Even though the field was 10 percent higher than for the case shown in Figure ,2S, the electrons inside 2.75,mm all passed. Table XI shows the parallel and perpendicular energies of the three trajectorie'» s which passed the reversal. For devices in which 0 it is 0 important to reduce the axial velocity spread, it would probably be necessary h 3 either to convert less of the total energy to transverse energy or to start with 0 ~ 0 a very thin beam. u , ^ _„. 0 • 6-29 0 . £f Q 0.9 ^ 1.0 1.2 Axial Distance, c:.n Figure 28. Electron Trajectories Through Simple Magnetic Reversal L poTnfuNC! senri LnuHCt-i£R ac-e , i : auc' ?z l. 1 •— Axial Figure 29. Electron Trajectories Thrc 2.0 2.5 o Axial Distance, cm <> 0 Figure 29. Electron Trajectories Through Compensated Magnetic Reversal o O & BLAN0 K PAGE TABLE XI ' Parallel and Perpendicular Energy;;for Rotating Beam h •v at Z = 4 mm past0 the reversal ^ , / " " " (kV) " " ' V„ (kV) Radius (mm) -t v 11 ^ ' o " " 2.5 166.0 " 33.7 2.6 179.4 , " 20.4 ^ 6.5 . GYROJRON INTERACTION THEORY " x i 6.5.1 Introduction and Summary Most of the literature on cyclotron-resonance masers-pr gyrotronsf deals with the interaction of the electrons with the electromagnetic fields of a . cavity by considering the electron beam as an active mcdiulu consisting of » 10 37 of'the relativistic dependence of the period of each upon its energy ' It has 0 ' , , {i o been suggested that the total energy of an ensemble 'of such oscillators,, initially " •'" a Or, ' i? ! having equal energiess which enter a radio-frequency field unphased will, in 38 aggregate. 1 , giveo u p energy to the field- . In this treatm%nt it is pointed out that while some'oscillators gain energy, some oscillators lose energy, and each - may periodically return to its original injection energy, the periods of cthe h 11 0 . " ' ,, "'JP energy cycle for all injection phases form a continuum, and the energy of the ; ' Q ensembl" ^ e need never return " to the initia-c l value. It " is recognized,.however" , " - ° % that the energy of the ensemble has minima and maxima as a function of time," - .; - 0 and that the optimum efficiency of a gyrotron is achieved when the .time of ~ 01 transit of the electrons through tiie microwave cavity corresponds to'an energy i« minimum of the electronic oscillator ensemble39'40 . , (v, <• " " I ° it 1 1 f-It is also recognized in the literature that; if the oscillators are * phased before they enter a region of high, radio-frequency fields (i.e., all electrons reach the high field'^region in a phase whichfgives up'energy to the "" " A" field), it is possible "to^achieve much higher efficiencies in gyrotrons or - °: f> ft 6-33 " „ o, • "' ** ' s- , " o „• ; - a •6 A • ft- •• if ' 0 , r , 0 •• _ ^ if- gyroamplifiers41' 421 43 (gyroklystrons) than is possible when the radio frequency field in which the electronic oscillators find themselves is spacially constant. For example, when the electron beam passes axially through a TE^ ^ ^ mode cavity, the orbiting electrons initially see only a low-intensity radio frequency electric field associated with the sinusoidal length variation. Not until the beam reaches the center of the cavity do the electrons find themselves in an „ rf field of maximum strength. At this point it is possible for them to have become phased. Zaytsev26 has demonstrated an efficiency of 31 percent for this configuration at 2.78 mm, and he references Russian calculations which predict as much as 60 percent efficiency based upon the transverse energy of the beam4"1. Another way of phasing the electronic oscillators in a low-field region while performing the energy extraction function in a high-field region is to build a multicavity device or gyroklystron in which the phasing function is performed in the input cavities and the energy extraction is done in an output cavity. Detailed calculations of the efficiency of such devices have been h made by following a number of electrons through the device on a computer, and efficiencies as high as 85 percent (with respect to transverse beam energy) have been predicted41. There is also a paper which proposes that the starting , current for oscillations in the output cavity of a gyroklystron can be increased while efficiency is improved by shaping the cavity in order to modify the axial variation of the azimuthal component of electric field42. Kisel'27 has demonstrated 43 percent efficiency in a gyrotron oscillator operating at the second harmonic of cyclotron frequency using this cavity-shaping technique (i.e., making the diameter smaller at the cathode end). The Russian theory is based almost entirely upon the numerical calculation of the energy exchange between electrons following helical paths and the microwave fields of cavities. The energy exchange for electrons of all phases is averaged to determine the performance of the device. The efficiency calculated depends not only on the axial variation of the oscillating fields and 6-34 i the time the electrons remain in these fields, but also on the relation between the strength of the fields and the transverse energy of the electrons, and on the difference between the cyclotron frequency. The theory also deals with interaction of electrons with fields at harmonics of the cyclotron frequency. 2 ,-,2 A simplifying assumption of v" ~\(S\ « 1 is usually made. This theory c will be reviewed in a later section of this report; it is quite analogous to the large signal klystron output gap calculations done by Feenberg45. While we haye found occasional reference to a "bunching para- meter"'13 and to space-charge debunching effects46 , in the Russian literature we have not found any well-developed small-signal theory. Because most of 0 .7. what goes on up to the output circuit of a klystron type amplifier can be very V 0' accurately described by a small-signal theory, we believe that optimization of the cavities and drift spaces in a gyroklystron can best be understood if small-signal quantities analogous to the beam-coupling coefficient and beam- loading coefficient are derived and a ballistic bunching theory is developed. We have made a start on this, and in addition to the previously-referred to review of the Russian theoretical work, this is also presented in a subsequent section of this report. We also present a section showing a method of deriving the small-signal parameters from energy-exchange computer calculations made using the equations of motion of the electrons. At Varian Associates, Jory and Trivelpiece developed a computer program which does calculate such energy interchange between cavity fields and electrons using the relati- vistic equations of motion. This program has been used to make calculations analogous to the Russion gyrotron efficiency calculations. It also can be used to arrive at values of small-signal parameters by the method mentioned above. This program is described later. 6.5.2 Review of Russian Theoretical Work In several Russian papers39 42 , the equations of motion of electrons following helical trajectories are solved by assuming that the cumula- tive interaction electromagnetic fields produce only slow variations in the 6-35 velocity and radius of the electron. Starting from the relativistic equations of motions in rectangular coordinates for electrons in time-varying electro- magnetic fields, solutions are found which have the form x + jy = -X + jY + r exp j (fir + 4>) x + jy = v + jv = jv exp j (nr + >p) Ay L r !] =v, z = v (28) by imposing the additional condition X + jY + (r + jri/i) exp (J2t + tp) = 0 . In the above equations, x and y are the coordinates of the electron in the direction transverse to the steady magnetic field and z is the coordinate in the direction of the field and the beam. X and Y are coordinates of the guiding center and r, v., v and tp are running values of electron helix radius, transverse t z velocity, parallel velocity and phase with respect to J2t. r is time, and -Si is the relativistic cyclotron frequency at t = 0 given by n = n m„ jr. i. V„ Vf V /3 = — and /3 „ = — are initial values. 8 = — and /3 = —z , which will be used J- c , " c tc zc later, are running values. The values of v , v and tp can be found from the equations v Z : - a (1 -(3*) sin (fiT + ip)+ a (1 ) cos (Qt + tp) - a^^ 2 2 vt = x Jl -B -? 2 (0+ ip) vfc = - a^ cos (fir + tp) - sin (S2r+tp) + £2QVt Jl -(3 - /3 v = sin {hr + il>) - ayS^ cos (Sir+tp) + a (1 ) (29) z z X 6-36 in which a =— (E + v x B). Examination of these equations shows that an o electron, phased so it sees a tangential, transverse electric field twice during each rotation, gains or loses energy. For an electron phased so it sees a normal, transverse electric field twice each rotation, the guiding center of the orbit assumes a circular motion which creates forces that oppose the acceleration of the electric field and so there is no net change in perpendicular energy. Zhurakhovskiy39 deals with the case in which E and B^ constitute the radio frequency fields of a plane wave propagating in the z direction. The expressions that are obtained by inserting the expressions for the fields in equations 29 are then averaged over a cycle of cyclotron frequency and expressions ' a.r,e obtained which may be used to calculate the energy of individual electrons after some interaction time with the field. With suitable summing over all phases, the efficiency of oscillators having this field configuration can be found. Rapoport, Nemak and Zhurakhovskiy40 deal with the interaction of electrons following helical paths with the fields of a standing wave having field components E = E sin ky sm wt / x o " • E B =—— cos ky cos cot (30) z c This is a particularly interesting case because,.in addition to providing a region in which the value of E is approximately constant (ky = tt/2) and there ^ is strong interaction between the electrons and the wave at frequencies which are odd harmonics of the cyclotron frequency, it also provides a region in which 3E /9y is maximum (ky = 0) and there is strong interaction between the X electrons and the wave at frequencies which are even harmonics of the cyclotron frequency. 6-37 Rapoport et al40 combine equations 29 and 30 which gives ex- pressions containing sine functions having arguments which are sine functions. Asa result, the averaging of these functions over one cycle of cyclotron frequency results in expressions containing Bessel functions of the first kind —«• 9 having the argument n/3 where n = oj/Q. For ("<<1, the expressions are simplified by representing the Bessel functions by the first term of the series expansion for the function of the lowest order, and then suitably normalizing. The resulting equations which describe the motions of the electrons are s dp . n-1 cos 5T = sm^)| cW . n-2 . 1 2 — = n sin ^p sm 0+o-+ — (p -1t) v in which n 77 E m n o o _ n-2 T = 0 clit n . coj J- 20 ni 0" = oj-nfl n r) E u> n oo^ n-2 " i 9n « " ccd ^ n- c. n 77 E . u- 1 = n o o ' R n-2 2w 0n , coj 2 2 ni nfi/3 Jb ^ e VQ = , H = 1 0 = (a; - nft) r + cut - nii . o ' Rapoport et al then approximate a one-half wavelength thick, strip beam by solving the equations using eight increments of £ from 0 to it together with sixteen increments of the, initial value of 6 to sum over electrons of variouu s initial phases. They present curves showing efficiency as a function of the field parameters 1 for n = w/fi = 1, 2 and 3. Their results show efficiencies which decrease as w/fi is a increased with peak efficiencies of 31%, 22%, and 17% for co/Q = 1, 2 and 3 respectively. The peak occurs at successively lower values of v ^ as u/fI is^ increased. However, if one makes a plot of the unnormalized field strength at'which maximum efficiency occurs as a function of for the varib.y.s rauo.s of w/n, as is shown in Figure 30, it is clear that the lowest fields exist for oj/fi = 1. Rapoport et al also give curves, as functions of u \ showing the optimum slip between the operating frequency and the cyclotron frequency harmonic and the optimum number of cycles for an electron to remain in the field to obtain the maximum efficiency for that value of v \ In other words, efficiency is a function of electric field, slip and interaction time and/fit is necessary to optimize all three independent variables to obtain optimum efficiency. u o "C"':, , Unfortunately, Rapoport does not give a set of curves for a thin beam optimized in position to give maximum efficiency. He does state, however, that the optimum efficiencies are 42%, 29%, and 22% for oj/S7 == 1, 2, and 3 respectively in the thin beam case. Also, the Rapoport calculations do not \\ apply to the practical case in which we are interested, i.e., in which the electric field component transverse to the beam varies sinusoidally in the direction of the beam as would occur when the beam passes through a cavity. Zaytsev in his paper, which summarized experimental results on gyrotron oscillators using such a field geometry, cites references giving theoretical perpendicular efficiencies as high as BO^o44. (Unfortunately, though w? have contacts with many libraries, we have not yet obtained^the reference.) Whi! 3 Rapoport's work seems to indicate that harmonic gyrotrons are not as efficient as fundamental mode gyrotrons, this does not seem consistent with, the experi- mental results of Kisel' et al. It may be that the decrease in efficiency-as gj/£2 is increased is associated with the assumption of Rapoport that E does not vary with z. This assumption is certainly not satisfied in the device built by Kisel'. In fact, Kisel' shapes the magnitude of the transverse fieldo very carefully. ^ 6-39 1,000,000 0.01 0.1 1.0 Or .8• JL Figure 30. Electric Field for Optimum Efficiency for Oscillator Analyzed by Rapoport, et al. ^ ' 1 ' 6-40. • r * ... \ o An analogous theoretical treatment has been applied to three- resonator, gyroresonant amplifiers by Demidovich41 et al and Kurayev42 et al. Demidovich does not appear to isolate groups of variables according to their effects upon certain performance parameters (i.e., efficiency, gain, start oscillation current, etc.) tiut rather treats the efficiency,as a function of all the variables simultaneously and'uses computational techniques to reach'the,., optimum efficiency. His best results indicate an efficiency of 85%. Space- charge effects are ignored. ' This calculation is similar in method and result to ballistic calculations on three-resonator klystrons which also ignore space- charge and yield efficiencies of the same general magnitude because the bunch .V; of electrons is squeezed by the fields oOthe middle resonator45 . o Kurayev has investigated the effect on efficiency and start oscillation current of changing the axial variation of the transverse electric field of a TE„ , mode by varying the cavity radius. 0ml Space-charge debunching effects in drift regions has been con| sidered by Kovalev et al46 . He ^calculates that, in'drift tubes beyond cutoff for TE„ and TR'l. modes, space charge effects can be neglected if Om Om ^ o < a S0.1 ' . (3 'Vi[ \ r I . 2 <|(ll\ o/ 4tt fi^ q in which I is the beam current in amperes, V is the parallel beam energy ' o 'i ..il in volts, r is the^beam radius in the same units as wavelength \,r}=Nn /e o v x 1 o o J v = 377 ohms, j3^ = — and q = x/vu . We wish to examine the applicability of this result further, because it seems to us that, in a gyroklystron, the drift spaces will be'very short, the cavities wiil be very long and, as in a traveling- wave tube, the space-charge forces will act in the'same region in which the bunching forces act; that is, in the cavities, where the modes'excited by the space-charge fields are not cut off. 6-41 6.5.3 Analogies Between Klystrons and Gyro klystrons There are obvious analogies between cyclotron-resonance devices and more conventional linear-beam devices. For this reason it is useful to study these analogies so that the body of information which has been it developed on linear-beam tubes can be used to understand gyrodevices. In this section we will show how a ballistic bunching theory very analogous to the klystron theory of Webster47 can be developed for gyroklystrons. A coupling coefficient'for TE^^^modes to fundamental mode gyrotron beams has also been developed. An' expression for the beam-loading coefficient has not yet been derived, but should be, because this will be necessary not only to determine the gain of multicavity devices but also to determine start oscil- lation conditions. Ballistic Bunching in a Gyroklystron (j In a cylindrical coordinate system (r, (p, z) we postulate a <> hollow-beam concentric with the axis of the coordinate system which passes a by two gaps which are infinitesimal in the 6 direction and one electron-path helix-pitch long in the z direction. The two gaps are separated by an angle 0 which represents the total angular progress of an electron around the axis in 0 going from the first gap to the second, and may be any number of revolutions. o In this system, each electron passes each gap only once, under small-signal conditions, so that 1=1 = I , the dc beam current. V v At the first gap, electrons undergo an energy change due to a sinusoidal voltage which exists across this gap. This produces, through a change in the relativistic mass of the electron, m = ym^, a change in the cyclotron frequency, 1 • gB = 6 = (33) n 7m O a 6-42 Asa function of time at the first gap the cyclotron frequency is fi-^) = 4>{\) = (1 - X sm wt) P. (43) A minus sign is used "in front of the sine term because a positive gap voltage decreases the cyclotron frequency, and we will later define a coupling co- efficient which relates the magnitude to E to the magnitude of M>m/S. Continuing in a fashion exactly analogous to that of Webster, the time of arrival, t , of an electron at the second gap can be found from , , A(f>m . 1 + —7— Sill Wt, (35) . ^'Multiplication by co give^ U)0 CO0 cot = wt + — + — -7— sin cot 2 1 Q. n = wt^ + — +' a sin cot^ (36) ,r in which ^ 9 = ndAcp/ It is possible to determine the current at the second gap from the expression (37) '(y 2-/ 5dt„ ij w This current may also be expressed as a seriesv1of harmonics 00 i (tj = Re ejWt2 (38) m Em=o . 6-20 in which the Fourier coefficients are given by U (T -jmojt9 A ) e " d(oit0) M, 0, 7T -jmcutr d(wt9) e 5 (39) -7T L w // For reasons which are not entirely obvious but which are well documented43 , this expression reduces to -jmojt m ^ fi 7T J u TT -jm^'t.^ + n0 + X sin wt^ ^ d(ajtx) YA-7T , „ -imn0 _ , __ = (-1) 2 It e ' J (mX) (40) y o , m\. // We see that the above expression is almost identical to the klystron expression48 except for the term (-l)m, which indicates that the bunch forms about the electron which passes the first gap while the voltage is going from accelerating to retarding, rather than the opposite as is the case in klystrons. This ex- f pression is probably more useful in seeing the similarities and differences between klystrons and gyroklystrons than it is for computation; because in a practical gyroklystror:N, the bunching 'cavity fields are distributed over some length and, to determine the current at some£ point on the beam, it would be necessary to sum the current, contributions due to all prior infinitesimal bunching gaps as was done by Tore Wessel-Berg49 in his analysis of extended-interaction klystrons. 6-44 Coupling Coefficient It is possible to derive a coupling coefficient betwee'n the TE^ ^ ^ modes of a cavity and a fundamental mode, gyrotron beam which passes through }) 1 " the cavity. The? transverse electric field E (or E , if we are concerned with a „ x cj> • hollow beam in a TE„ , mode cavity) is taken as 0,n,I f-\ E^ = E^os kz (sin (wt (41) Because relativistic effects cause changes in the axial velocity, e CiJ ^ v , to occur when the transverse velocity,1 v., is changed by a transversa acceleration as described in equation 29, it is ,necessary to relate ^(b/ifi to ' // f) both Av, and Av and then determine what these velocity changes are in terms t 2 ' ' n of E to find the coupling coefficient. We, will ignore interaction with Av + Av (42) t 3v 2 Since D eB (43)' = n - Tv~m O 2v -1/2' z (44) 2 we can perform the indicated operations and obtain $ v v 2 A0/0 =-r (;r|Avt + -| Avz (45) From equations 29 in the prior section and equation 41 above, 6-45 e 2 v = ( E (1 - P ) cos (kz) sin (wt + sin (S2t + ip) . ymo» -e v = E (3 8 cos (kz) sin (wt + sin (fit + ip) . z ym i. L Z (46) For very small accelerations, we can assume that the transit time of electrons through a cavity will be constant and will be given by T = L/v in which L is z the cavity length. Also in the above expressions, k is set equal to tt/L, and z = v t. Using the above assumption, Av and Av can be found by integrating z t z v andiv between the limits of -T/2 and +T/2. Integrating equations 46 in c Z ^ this way and combining with equation 45 gives us + L/22V eE Ad)/. LAi Xco/ s (kz) ssin(w t + tf) sin,.(fit + ip) dt' . (47) 4> ym^c -L/2V ' After some manipulation, one obtains 'COS I cos j = A cos (^ - ip) .2 ? 2 5 cos £ cos g - cos (^ + Ip) — - K — + £ (48) L2 * 2 u. u r ok'L k'L 5 k'L , . w fl-co _ . . . m which £ - ——- , yC = —z— - —— , k' = — and o = . The constant 2 2 2 v z co eE^L V V/V t V L t n A = . 2 v V A. v 4(1+V/V ) 4ym c z z n o moC 6 if one lets V be the beam voltage and V = = 0.511 X 10 volts. n e 6-46 V The coupling coefficient is obtained for a particular value of ip by adjusting to maximize A 'cos g cos g cos g COS g A0 /<£ = A + —— (49) m tf IT 7T , 7T l~2~ Another case of interest in which some simplification is possible is that in which the electrons make many orbits within the cavity. In this case £ is large and the-maximum value of is given by cos 4 cos | A<£ /(p = A + (50) l-t -+£ 2 s 2 For this case, at synchronism, that is, £2 = w, £ = 0 V/ EA v. Vy . i "/ r 4 . 1 TL t n (51) m = * A = IT T ~ ,(1-V/ ) z " V n The most important fact revealed by equations 49, 50 and 51, other than the magnitude of the coupling coefficient and its variation with the degree of syn- chronism, is the fact that the coupling will vary depending upon the position of the beam scallops within a cavity. If the electron makes three or more orbits in a cavity, this variation due to scallop position is about ± 10% or less. Another form of equation 51 which is useful when considering output cavities can be derived by. differentiating equation 43 to get A£ = -Ay (52) 6-47 Noting that Ay = AV/V^, we can then put equation 51 into the form v AV = -;Eixi7 - °<53> z )! This expression gives the change in kinetic energy (or voltage) for an electron passing through the cavity with synchronous magnetic field applied. 6.5.4 Determination of Performance Parameters Using Numerical Methods In the Literature which was reviewed in the prior sections, also in the preceding section on small signal theory, it is a common practice to u eliminate from consideration those electromagnetic field components which make the mathematics difficult. Also, a weakly relativistic beam assumption (i.e., • ——12 bS « l)is frequently made. These assumptions are not always justified, and it is convenient to be able to check analytic expressions to see how well they represent the truth without resorting to experiment. It is also useful to be able to find parameters such as electronic admittances from numerical computations of electron energy changes when mathematical analysis becomes intractable. The approach is to use a Langrangian description of the electron motion, neglecting space-charge forces, but including relativistic effects and the complete resonator fields. Either equations 29 in section 6.5.2 or the program described in section 6.5.5 could be used. The neglect of space charge is not a very serious limitation, since the beam perveance used in these devices will be very low, most likely in the vicinity of 0.1 /uperv. The energy trans- s, ferred from the resonators to the beam for given resonator voltages is found by computing the change in the kinetic energy in the beam after each resonator. By performing this analysis for a sufficient number of combinations of resonator voltages,, it is possible to determine the real part of the beam loading of each oi the resonators of the multiresonator amplifier, and also to determine the real and imaginary parts of the transfer admittances. These parameters, 6-48 together with computed values of R /Q and loaded Q factors of the resonators, sh i are the necessary parameters to determine stability and gain in the amplifier. Of course, a consistent d^tnition of cavity voltage must be used for Rg^/Q all admittances. Computations 01 the efficiency and'effects of non-ideal conditions of the dc beam - for example, axial velocity spread and finite beam thickness - can be simulated using the same approach. r/ The novel part of the analytical method is to establish the relations between the beam loading and transfer admittances from the informa- tion of the kinetic energy in the beam. The derivation of these relations is done for the simplest case, a two-resonator amplifier, in the next subsection. The expression for gain and the stability conditions are given in the subsection following that. Beam Loading and Transfer Admittance The power flow in the two-resonator amplifier used to demonstrate the method of determining the beam loading and transfer admittance is shown in Figure 31. o 01 Resonator Po + AP02 No. 1 V1 Figure 31. Model Showing Power Flow in the Amplifier 6-49 The power flowing from the resonator to the beam is indicated ,, by P and P9. These powers represent the beam-to-circuit interaction only and do not include any circuit losses. The overall increase in the kinetic energy of the beam Pq is indicated by AP^ and ^Pq/after each resonator. (V A negative value of these quantities means the energy has been transferred from the beam to the circuit. The resonator voltages V^ and V^ are calculated on the basis of the dominant electrical circuit field, and can have any one of > many definitions as long as it is consistent with the voltage used in computing the (R ^/Q) of the resonator. Complex voltage amplitudes are used in the following to demonstrate the general validity of the theory. The symbols used for the complex electronic admittance of the resonators are Y , and Y . The transfer admittance between the two resona- el e2 tors is Y^9. From the indicated power flow pattern in Figure 31, we get the following two equations: P = Re Y V V *1 = Ap (54) 1 12 el 11 J 01 P2 = Re R Y12 VlV + i Ye2 V2*} = ^02^01 <55> Since V V * and V V * are always real, we can express equations 54 and 55 JL X 0 by: " ' - G IV I2 = Ap'^ (56) 2 el 1 01 0 and Ee Y V + G V = AP -AP (57) -i { 12 lV} f e2 2 02 01 where G „ and G . are the'beam-loading conductances of the resonators, el e2 ^ From equation 56, we get: 2AP 01 (58) el V. Since V^ and V9 in general are complex quantities, we have Y (59) 12 12 V. V. (60) V, V. (61) From equations 57, 58, 59 and 60, we get: j/3 1,2 Y V. V, 12 + ¥Ge2 = AP -AP (62) 02 01 1 or Y V, V cos / B - (a -a )\ + - G V 2 12 1 2 1*12 v 2 ly 2 e2 = AP -AP (63) 02 01 A maximum energy transfer from the beam to the circuit is obtained when (64) h12 v 2 1; Having selected a (which could be a,=0), we can adjust a in the computations 1 X z until the kinetic energy in the beam is minimum. The angle fl is now determined (a) - a (65) 12 v 2' opt lj With G^ adjusted for optimum energy transfer to the second resonator, we get^from equations 63 and 64 6-51 1 1 Y, V V + G . = v(AP -AP ) (66) 2 12 1 - 2 2 e2 2 02 Ormin The computation can be repeated for a different value of . In this case, substituting the appropriate values into equation 66, we get 1 V, + i G V„ = v(AP '-AP ') . 2 12 V 2 e2 02 01 min (67) Subtracting both sides of equation 67 from equation 66, we get 1 1 Y V. V. o 12 vi' 9 12 (AP - AP ) - (AP ' - AP ') (68) ^ 02 Ormin 1 02 01 ' min ' giving o (AP -AP ) (AP »-AP v 02 OTmin 02.. 01 min (69) 12 V I V, ' V. 2 ( 1 I) It should be emphasized that this derivation is valid only for small signal conditions where Y , G and G can be considered constants (independent 1X&» c> v01 . " 02 • of the resonator voltages), The beam-loading conductance in the second resonator is found by substitution of into equation 66: 12 v 02 Ormin G (70) e2 V 2 c (Ivl'l - N) (AP -AP \ ly t| _ //^p i_/\p |V 1 • „ tl „ 02 Ormin Til ^ 02 *01 'min | l| (71) e2 - 4 V 2 V - 2| (| l' vi|) 6-52 The results of equations 5S, 65, 69 and 71 show that the beam- loading conductances and the transfer admittance can be found from the computed beam power after each cavity. The presented derivation was done to demonstrate this point with the example of the two-resonator amplifier. If we adopt the method, we will develop a'computation sequence that is more economical for the computer than outlined in'Athe present writeup where the emphasis has been placed upon simplicity to demonstrate the principle. For example, a variation of the second cavity voltage rather than the voltage of i ie first resonator results in a little more arithmetic derivation to find Y, „ and G but.would 1,2 e2 eliminate a repeat of the integration of the equation of motion through the first resonator. It should furthermore be a trivial matter to extend the procedure for analysis of tubes with more than two resonators. Power Gain and Stability The power gain in a device is normally defined as the ratio of the power P delivered to the external load of the output circuit and the available JLi power P. from the generator attached to the input circuit. The relation between the available power from the generator and the circuit voltage of the input resonator has been shown to be50 : o p =,I (-^-1 Q V V * Y Y * (72) i 8 \ Q ^ ext, 1 11 11 * K ' This equation is valid for any type of resonant circuit. RA In this equation (-r— is the characteristic impedance of the \QA input resonator, Q is the external Q factor and Y is the overall complex admittance of the input circuit including beam loading, circuit loss and the external load. We shall define the resonant frequency of the input circuit as the frequency at which the sum of the imaginary parts of the admittances entering into Y is zero. In this case: \ \ t\ 6-53 Y =G +G +G // (73) 1 el cl ext;l J v ! Avhere G , is the circuit loss conductance and G <> is the load conductance cl // ext, 1 jprese'^ec^by a matched input waveguide. / - The sum of G + G can be expressed in terms of the cold cl ext, 1 circuit Q factor Gcl+Gex«,l7IT7r (74) where 1 _ _L_ 1 , (75) QTL1l Q„c,l Qqext f , .1 Q is the total Q factor of the input circuit, ,, -Lil Q , is the Q factor associated with,the circuit loss, .cl ^ Q , , is the external Q factor, ext, 1 If external load of the output resonator is matched to the output transmission line, the power dissipated in the external load is given by: V V * p L2=i 7|rf- UL *ext,2 From equations 72 and 76, we get the power gain. - V V * 1 2 2 2 - Q \QJ2 ext, 2 Gain = s (I) "„t.i ww V V * 4 2 2 or Gain = TTT r— (77) /—\ —) Q Q Y Y * V V * \Q/2 \Qjl eXt2 6Xtl 11 11 6-54 V 2 1 * —* The ratio of the resonator voltages rr- is found from the power balance in the putput circuit: V 1 II 1 — V V * Y = Y •' V V * (78) 2 22 2 2 12 12 therefore, V. 12 (79) V. From equations 74 and 79, we get Y Y * 4r 12 12 Gain = (80) Y Y * Y Y * Q Q (Q\ (QL ext2 extx 11 2 2 The transfer admittance Y and the real part of the beam-loading admittance entering into the expressions for Y and were determined above. The imaginary part of the beam-loading admittances are not known. These susceptances can be compensated for by a slight detuning of the resonators. We obtain this condition by defining the resonant frequency of the circuit as the frequency at which the sum of the imaginary parts^of the admittances entering into Y^ and Y9 is zero. This does not result in any practical limitation.of the theory, since the beam-loading susceptances normally are changing very slowly with frequency and we are concerned here with narrow- o band devices. With this assumption, we get (81) 6-55 and J 1 + 2joQ L,2 + G (82) ei-2 a In equations 81 and 82, CD is the loaded value of the resonator, including L, n the internal losses, and o is the frequency tuning parameter OJ-OJ o = (83) LO where u> is the operating frequency and cu is the resonant frequency. We ° .J have now determined all parameters necessary for computation of the gain. We shall next state the condition for stability. The nth resonators are stable against oscillations provided that 0 G < (84),. e, n , /n QL'n This condition can easily be checked, using the value of G calculated for 6 ^ n the various resonator modes from the method outlined previously. c. A 6.5.5 Ballistic Trajectory Analysis Code Several years ago, a proprietary computer code was developed at Varian to calculate the motion of char'ged particles in electromagnetic fields with an applied dc magnetic field51 . The relativistic force equation for a particle having mass m, charge q, position r, and velocity v in an electric field E and magnetic field B is (mks units) 6-56 (d/dt) [mv(r,t)] = [ E(r, t)+v(r, t)xB(r, t)] . (85) cf'> - By writing the relativistic mass in terms of rest mass and velocity 9 1/2 m=m /(l-v"/c ) , ' (86) o o and by performing considerable algebraic manipulation, we can put equation 85 into the following form: 2 (d/dt)v(r,t) =(q/ymQ) {E(r,t)+V(r, t)xB(rf t) -[7(7,t)/c ] [7(7, t) • E (7, t) ]} . (87) This form is more convenient for numerical integration. The rest mass is denoted by m and the relativistic factor 2 2 l/2 0 — l/(l-v~Vc ) " is denoted by y. The position of the particle is given by r, where iTi. ° (d/dt)r = v(r, t) . „ (88) (> O The code' operates by performing a numerical time-stepping integration of equation 87 ijx rectangular coordinates. The time step is typically" U 6 chosen to be small enough that the error per step is less than one part in 10 . « The code is designed to use either TE or TM cylindrical waveguide or cavity- fields. All E and B components for the particular mode being considered are included in the calculation. In addition to the electromatic fields, a static dc magnetic field can be included in the calculation in the form U 0 B^ =-BoaI1(27r r/L) sin (2?r z/L) , ">(89). B = B- [ 1- a I (2?r r/L) cos (2TT z/L)] ; - (90) Z ° 0 " „ " " where I and I, are modified Bessel functions, ci is an amplitude parameter, O 1 %'s • » ... i? ''' and L is a length parameter. These parameters can be chosen to apply a ^ a.b 0 t, ••)!: O ' - 6-57 magnetic mirror field with mirrors located at z==L/2, and with a mirror ratio, R = (l+Q)/(l-o;).. Or, \vitha=0, a uniform dc magnetic field'is applied. m This code can be used to calculate the motion of electrons through a gyrotron cavity including all cavity field components, all relativistic effects, and any nonlinear effects in the motion. -It is, of course, necessary to specify o ' O an amplitude for the cavity fields which is consistent with the cavity loaded Q and the energy transferred from the electrons moving through the cavity. There- fore, to simulate an interaction involving specified beam and cavity parameters where nonlinear effects exist, it will in general require iterations on the assumed field amplitude until a self-consistent calculation is obtained. ci> ' « A limitation on the code is that;,it does not include space-charge effects. This does not appear to be a serious limitation for gyrotron devices. o 0 // u o o a 1) Ci 6-58 7. CHOICE OF APPROACH y • Iii this section we will summarize the comparisons which have led to the 1 n ' r choice of an optimum approach to generate 100 to 200 kW cw power at 120 GHz. i t We have considered published power'achievements, the adequacy of models for calculating performance11 o , calculated performance where possible, and the degree of risk in solving required technical problems. 4 „ 7 ° U The generation of 12 kW cw with the gyrotron oscillator at 107 GHz represents significantly more average power than is claimed for any other approach. The gyrotron efficiency of 31% is also much higher than that achieved by other approaches at that frequency, and is comparable to linear-beam efficiencies at lower frequencies. Possible drawbacks of the gyrotron are the limited experience with it in the United States, the difficulty of.generating the required beam with high i. transverse velocity and low velocity spread, and the inherent stability problems in building an amplifier. v> The linear-beam amplifier has received extensive development effort at t> & lower frequencies. Its performance is probably the most predictable of all the i approaches considered. However, it encounters severe power density problems at the desired frequency and power level. To bring the power density within ; allowabl'K'-vge limits and yet maintain reasonable efficiency, we hav< e concluded tha•t an=^ output circuit having tapered impedance is necessary. The tapered impedance circuit represents a new development involving the risk that new problems will be encountered. Furthermore, even if the proper circuit is developed, the calculated performance is " still marginal with respect to efficiency (10 to 15%) "and power density 5 kW/cm 2 )': o The beam problems for the linear-beam approach are judged to be similar in difficulty to those for other approaches. The required current density 2 (700 to 1000 A/cm ) is higher than that required for other approaches, but the beam is a conventional one with no requirements for high transverse velocity or periodicity. 7-1 Power levels achieved in linear-beam deviccs near 100 GHz have been limited to the 100 W level, which is considerably less than the 12 kW gyrotron. In making that comparison, it must be noted that the previous lack of demand for high-power linear-beam devices is a factor to be considered. Another disadvantage of the linear-beam approach is the complexity and relative difficulty of fabrication of the interaction circuit. The delicate nature of the linear-bcarn circuit will increase cost and decrease reliability. A consider- ation of all these factors has led to the conclusion that the linear-beam amplifier is not .the optimum approach. In comparing the gyrotron to the Ubitron, it is concluded that the allowable beam area and the circuit design considerations are similar. The main problem with the Ubitron appears to be placing the periodic-magnetic structure sufficiently close to the beam region. At short wavelengths, this requires making the magnetic structure either inside or part of the vacuum envelope. Even when this is done, there is an inherent variation in the strength of magnetic field with radius. It is concluded that this variation with radius is very likely responsible for the relatively-low efficiency for experimental Ubitrons; if so this would appear to be a fundamental disadvantage. The periodic-beam TM interact!^,, of the Ubitron, compared to that of . the gyrotron, has slightly less beam area but similar circuit dissipation density. Analytical models to predict performance are less developed than for any of the other approaches. An inherent disadvantage may be that the effective E field z is small compared to the peak E field, but the E field may not be the only z z important field in the interaction. An advantage of the TM interaction is that lesser amounts of transverse velocity are required for the beam. This makes the beam generation problem easier. Mainly because of the lack of analytical models, we consider the TM periodic beam approach to be more risky at this time. A reconsideration of it may be useful in the future. 7-2 In summary, it is concluded that the gyrotron is the optimum approach to pursue. It is recognized that generation of the required beam is one of the key problems which will need considerable effort. Stability with respect to oscillation is another problem area (which is common to all approaches having large beam area). The main areas of risk with the gyrotron are efficiency and stability. However, the values of efficiency already demonstrated are high enough to be very encouraging. 6-3 8. DESIGN CALCULATIONS FOR THE PREFERRED APPROACH 8.1 GENERAL DESCRIPTION A preliminary amplifier configuration is indicated in Figure 32. The essential elements of the amplifier are an electron gun, the microwave inter- action circuits, the collector, the focusing solenoid, the water cooling system, and the input arid output guides and windows. The electron gun shown is the magnetron injection type having a hollow cathode. The microwave circuit is shown as three resonant cavities which will be designed to operate probably using the cylindrical mode for the output cavity and for the other two. The only desired energy coupling between cavities is'by the electron beam. The, desired beam is a hollow beam with all electrons having an equally- large fraction of their energy in transverse velocity. The beam is adiabatically compressed while entering the main solenoid to increase transverse velocity and to get area convergence. The dashed lines in the figure indicate the average position or guiding center of the beam electrons. The design which optimizes efficiency requires a magnet field near the fundamental cyclotron resonance value. This requires a cryogenic solenoid at » 120 GHz. Details of the solenoid system are omitted from the figure. The microwave operatior^of the device is essentially what has been called ' f'< ( a "gyroklystron" 42 . The input'signal is applied through a conventional waveguide // and vacuum window to the first cavity. This cavity applies a microwave energy modulation to the beam, primarily in the form of angular velocity modulation. This velocity modulatioivyartially converts to angular density modulation, which excites microwave fields in the second cavity. The second cavity further modulates the beam to maximize the angular density modulation in the third or output cavity. The output cavity is designed so that the induced fields extract a maximum fraction 8-x . / BLANK PAGE Figure 32. 100 kW, 120 GHz Gyroklystron of the transverse velocity of the electrons. Output power is removed using oversized waveguide and an oversized output window. Water-cooling channels are shown schematically on the collector, the cavities, and the output waveguide. It is expected that heat flux densities in 2 all these areas will be about 1000 W/cm . It is expected that the collector will be insulated from ground to allow measurement of beam transmission. 8.2 MICROWAVE CIRCUIT DESIGN The choice of a multicavity gyroklystron configuration, as opposed to a purely traveling-wave cyclotron resonance amplifier, is based on linear-beam tube experience where the TWT is more difficult to stabilize against spurious oscillations. In this application, also, the broad bandwidth of the TWT is not needed. The gyro klystron does retain, to a degree, the distributed nature of the TWT by the use of cavities many wavelengths long. This is advantageous with respect to field breakdown and power density. The choice of a multicavity amplifier as opposed to a single-cavity oscillator should result in significantly higher efficiency. The inherent advantage o of the multicavity system is the ability to separate the bunching cavities from the output cavity, allowing each to be optimized for its particular function. • _ The choice of a particular cavity mode is based primarily on the amount of useful electric field that can be achieved in the cavity for a given power density of ohmic losses in the cavity walls. At millimeter wavelengths, power density is a major consideration. Referring to Figure 22 which shows the figure of merit for TE„ , cavities, 0ml if we pick a cavity mode, a value of D/L, and a peak electric field value, then the average power density in the cavity can be determined for any frequency. -7 Putting in numbers for 120 GHz, we have 6 = 1.92 x 10 m for ideal copper and = 2.5 mm. For a figure of merit of 1000 from Figure 22 for a TE cavity 7 with D/L = 0.4. and for a peak E , value of 10 V/m, we calculate an average 2 * power density of 385 W/cm . In a practical device at 2.5 mm wavelength, the actual Q might be as low as one-half the ideal value because of surface effects. ° 2 This would double the average power density of 770 W/cm . The peak power density at the center of the cavity will be twice the average value by virtue of the sinusoidal variation of field with the axial coordinate. Therefore, the peak 7 value for E^ of 10 V/m appears to be acceptable with rcspect to power loss density. This is similar to the field strength used in the Zaytsev experiments and is representative of the field strength needed for high efficiency. An obvious requirement for high efficiency is that the electrons, in passing through the output cavity, lose as much of their kinetic energy as possible. Equation 53, which gives the energy change for electrons in long cavities with synchronous magnetic field, is repeated here for convenience: v AV = - - E ,\ 7 v1- • (53) 7T i1 A •z The energy change given by this equation should be comparable to the transverse voltage equivalent of the electrons or somewhat greater to account for lack of synchronism. Increasing the ratio v^/v^ results in increased energy transfer, but also results in more space-charge depression of beam potential. Velocity spread effects become more critical also as the v7U v Z ratio iis increased• . A practical value based on the Russian experience is vVv = 2. £ Z If we use an 80 kV beam and require AV = 80 kV to allow for clesynchronous 7 effects, then with E = 10 V/m and v /v = 2, the required length is L = 12.5 mm -L t Z or 5 wavelengths. One could reduce the velocity ratio and increase L for an equivalent AV. One concern in doing this is that the cavity is already operating- very close to the cutoff condition, as indicated by the equation 2 f/f =N/i-(V2L) (91) where f is the operating frequency and f is the cutoff frequency. For L = 5 A. 8-4 we get f /f = 0. 995. Therefore, a small change in cavity diameter from one end of the cavity to the other can effect the field shape significantly. One must there- fore put tight dimensional controls on the cavity. Another concern in increasing the cavity length is that the numl&r of possible resonances increases. This could make stability problems more difficult. The Russian references indicate cavity lengths in the range of 4 to 7 A. A choice of 5 A appears reasonable by comparison. It is noted that the ratio L/v is the transit time of the electron through the !) z cavity. /'Therefore, tradeoffs between length and velocity ratio for a given value of AV do not change the transit time through the cavity or the number of orbits. One would expect the number of orbits to be important in determining beam loading and therefore to be a measure of stability, i.e., a shorter transit time will result in better stability. Having chosen a cavity length of 5 /\, we should recheck the effect of this length on the value of the figure of merit in Figure 22. We now have a vaJue of D/L =0.45 compared to the value of 0.4 originally assumed. Reference to Figure 22 indicates essentially no change in the figure of merit. The Qq of a TEQml cavity was given by Equation 9. In this case, we calculate a Qq of 1.45 x 10 using the,ideal skin depth for copper at 120 GHz. In practice the Q might be as low as 7000 due to surface effects-. The required Q o ext (degree of coupling to the output waveguide) can be determined by calculating the total power loss in the output cavity and by making use of the proportionality between Q and power: o output 2 Based on the ideal copper average power loss clensitv of 385 W/cm , the it total loss in the output cavity is 980 W. If we assume that the actual losses will be double, this value, that still represents only 2% of the output power of 100 kW and the required Q is 300. In this case, the total Q, which is given by = ' (93) T o ext is essentially equal to.Q^^. I! The relatively-low value of Q means that the output cavity is very heavily loaded by the output guide. This is good with respect to efficiency but may create some problems with stability. The implication is that other imdesired cavity modes can also build up appreciable field strength with only, modest amounts of total power excitation. It will be necessary to load any resonance which can have inter- action with the beam to a low enough Q to avoid oscillation. Cold testing will need to be done to determine Q values, and stability calculations can then be made. Calculations of the performance of the selected output cavity were made using the computer code described in Section 6.5.5. Figures 33 and 34 show plots of energy versus time for electrons passing through the cavity with various entrance 5 phases. The calculations all use a peak electric field strength of 10 V/cm at a wavelength of 2.5 mm. Initial electron velocities are v = 0.45 c and v = 0.225 c t z corresponding to an 80 kV beam. The calculations are actually made for a TE^ resonance with the electrons orbiting around the guide axis. This is equivalent to the TEQ21 mode operation with the electron guiding center at a radius given by 2 TTi/A = 1.84. In each case, each electron enters the cavity at t = 0. Because v can change slightly due to interaction in the cavity, each electron may leave the z cavity at a slightly different time. However, these differences are typically less than a few percent, so there is negligible error in assuming that all electrons exit from the cavity at the same time. The transit time will also vary from case to case, depending on how much energy the electrons lose. .."v>. . The small-scale rippling of the energy versus time curves is caused by the time-varyin" g nature of the force"s on the electron. The E p, component of field ^oe° s to zero twice during each rx cycle. d-'J 00 I 140 cot Figure 33. Electron Energy vs Time with Synchronous Magnetic Field Encl of Cavity 130 140 Figure 34. Electron Energy vs Time with Magnetic Field Set for 4.5% Slip. Figure 33 shows the calculated energy for the case where the dc magnetic field is set at the synchronous value. For this case, the electrons which gain energy nearly balance those which lose energy, so that the average energy of all electrons remains essentially constant at any time. However, the electrons exit from the cavity with considerable energy modulation as indicated in the figure. A case foinvhich the magnetic field is 4.5% less than the synchronous ij value is shown in Figure 34. For this case, there is an average energy loss for all electrons of 30% for the complete transit through the cavity. When all phases of electrons are included, the calculation simulates the behavior of a single- cavity gyrotron oscillator. Assuming" the cavity had been appropriately loaded by an external load to allow the electric field to have the value specified in the calculation, the conversion efficiency of the oscillator would be 30%. The same calculation can be used to predict the performance of the output cavity of an'amplifier. In the amplifier, the beam entering the output cavity would be bunched so that electrons would enter with only certain phases. If we assume the electrons entering the output cavity are all contained in the phase interval 7R /4 to 5TT /4 (a TT-radian bunch length), then the average energy loss by the beam is 50%. A tighter bunch with electrons in the interval TT/2 to TT {it /2 radian bunch length) would result in 55% efficiency. In a linear-beam amplifier it is generally possible to achieve 1 to 2- radian bunch lengths. The bunch length that can be achieved in the gyrotron is not yet known. Velocity spread may have an important effect on bunch length in the gyrotron, but if reasonable bunch lengths can be achieved, amplifier conversion efficiency should be excellent. Additional work needs to be done to define the optimum configuration for the input and center cavities. The power levels in these cavities should.be c sufficiently reduced to allow the use of TE cavities. The cavities can be o 0 reasonably short because large changes in beam energy are not needed. With 3 a cavity length of 3 free space wavelengths, the ideal Q^ would be 7.9 x 10 . Further development of the small-signal gain analysis is needed to properly define the required cavity loaded Q values. It is estimated that loaded Q of about 1000 would be appropriate. IV Overall stability considerations may also effect the design of the input and center cavities. Waveguide modes such as the TE,, cylindrical mode 11 0 will propagate through the entire length of the device. It will be necessary to o measure by microwave cold tests the degree to which these modes will be excited, and to calculate the pow,er gain for "these modes with the beam present. The requirement to keep these modes below the oscillation threshold may require a reduction in the number of cavities or a modification in their configuration. Cold test measurements are needed also to check for imdesired resonances and to verify that the proper coupling can be obtained to the external waveguide to load both clesired and undesired modes. i; (t In checking the undesired modes, it will be necessary to consider all polarizations of linearly-polarized modes. It is believed that the type of coupling a ,, system ,shown in Figure 32 can accomplish the proper loading. If this type of loading is not adequate, the type of loading used by Zaytsev26 can be used in-which all modes, V including the desired mode, are loaded through the collector. A combination of the ft two types is also possible. o <5=- 8.3 ELECTRON GUN DESIGN The preferred approach for the electron gun design is a magnetron injection gun followed by adiabatic (magnetic) beam compression. This type of gun has been used in the reported Russian experiments. Its main limitation is the spread in velocity distribution which results from electrons from different regions of the V cathode receiving different amounts of transverse velocity. 8-10 ^ For the 120 GHz amplifier, the beam dimensions would be: average dia- meter 1.45 mm and radial thickness 0.4 mm. With a beam current of 4 A, 2 this resultscin a beam current density of 220 A/cm , which will require an area convergence of 100 with respect to the cathode area. " c No specific beam dimensions are given in any of the Russian references obtained so far. Neither cathode dimensions nor magnetic compression ratios ( \ are given. It will be necessary, therefore, to calculate what values will be " acceptable in practice. An existing proprietary Varian computer program has u the capability to analyze the required magnetron gun geometries with appropriate magnetic fields and relativistic forces. It is planned that calculations with this code will be used to optimize the gun design. ' i it A study should'be made of other approaches to obtaining the beam with high transverse energy which may be advantageous with respect to velocity Cv spread. These include the use of multiple magnetic perturbations, electro- static deflection, and rf deflection. 8.4 . FOCUSING SOLENOID i. The 120 GHz amplifier will "require the use of a superconducting solenoid. An axial magnetic field of 48, 000 g will be required with a uniformity of.about + 0.1% over a,volume 2 mm in diameter and 35 mm long. The solenoid should have a room temperature inner diameter of 3 inches. The length and outer 'i diameter of the solenoid are not critical. It is believed that such a solenoid is V ! within the state of the art. - In addition to the main solenoid, an additional room-0 temperature coil might be used surrounding the gun region to aid in controlling the beam compression. ^ " 1 ° u 8-11 8.5 COLLECTOR AND WATER COOLING DESIGN The collector will be a water-cooled copper cylindrical structure similar to that used on many high-power tube designs. A inner diameter of five inches and . 2 a length of nine inches would result in an average power density of 280 W/cm . This is a typical allowable average density. Details of the collector shape will be worked out by computer simulation using the actual beam parameters. The / 2 shape will be chosen to keep peak power densities below 1000 W/cm . Varian has considerable experience in the operation of collectors at the megawatt level. High-power beams in. linear-beam tubes are often subject to beam j' pinching or self focusing in the collector region'paused by partial ion neutralization. The gyrotron should,be less subject to this effectbecause of the high rotational velocities of the electrons, but x-ray sensing protective circuitry at the end of the collector may be employed if computer simulations indicate that pinching can occur. Water-cooling channels will be built into the tube body over its entire length to accomodate some beam interception as well as microwave losses. The output waveguide will be surrounded by a water-cooling jacket and cooling will probably be used on the guides connected to the input and center cavities. The cooling channels are indicated schematically in Figure 32. Details will tie o specified after the microwave design is finalized. 8.6 WAVEGUIDES AND WINDOWS These components were discussed in detail in earlier sections of this report. For the 100 kW, 120 GHz amplified output, an oversized output guide and window will be needed. The input guide and window will use conventional rectangular guides. In addition to'the input and output guides, it is expected that a guide would be coupled to the second cavity to monitor field strength and to accomplish some cavity loading. 8-12 8.7 SUMMARY OF DESIGN PARAMETERS A summary of the tentative design parameters for a 100 kW, 120 GHz amplifier is shown in Table XII. It should be emphasized that these values are tentative, since much design optimization remains to be done. 8-13 TABLE XH w 0 Tentative Design Parameters 120 GHz Power Output 100 kW Output Cavity TE Length 5 A.0 (AQ = free space wavelength) 12. 5 mm Diameter 5.6 mm 4 Ideal Q0 1.45 x 10 Expected Qrj, 600 Peak E<£ 105 V/cm Ideal Peak Loss Density 800 W/cm2 Input and Center Cavities TE Oil Length 3 XQ 7. o mm Diameter 3.1 mm 3 Ideal Q0 7.9 x 10 Expected Q-p 1000 Beam Voltage 80 kV Beam Current 4 A Average Beam Radius 0.73 mm Transverse Velocity 0.45 c Axial Velocity 0.225 c Electron Orbit Radius 0.18 mm Outer Beam Radius 0.93 mm Inner Beam Radius 0.53 mm Current Density 220 A/cm2 Drift Tube Radius 1.25. mm' Solenoid Field Strength 48,000 g I. D. of Solenoid 75 mm Diameter of Critical Region 2 mm Axial Length of Critical Region 35 mm o 8-14 9. CONCLUSIONS The study program has led to the conclusion that while a number of device appi-oaches are probably capable of producing 100 kW cw power at 120 GHz, the gyrotron type of interaction offers advantages at 100 kW and is the best prospect for still higher power levels. The linear-beam approach would require develop- ment of some unproven circuit impedance tapering and, even then, would, result in critical circuit fabrication and cooling problems. The cyclotron resonance devices (gyrotron) operating either at the fundamental or at a harmonic of the cyclotron frequency appear to offer advantages over other periodic-beam devices. These advantages include larger beam cross section and demonstrated power and efficiency capability. The analytical models to predict the performance of the gyrotron ai^l the periodic-beam devices are not fully developed. In this study, we have attempted to summarize the published analytical work and to" develop additional analysis in some areas related to the gyrotron. One of the important aspects of the gyrotron device that needs more development is the means of generating the required electron beam. A high transverse velocity with minimum velocity spread is required. The magnetron injection gun approach that has been used in the experimental gyrotrons has resulted in excellent (30 to 40%) device efficiencies; however, there are indi- cations that the beam velocity spread is a determining factor on efficiency at the present levels. Another aspect of the gyrotron that needs further investigation is stability of the device as a multicavity amplifier. For the desired output power, it will be necessary to use beam tunnels that are large in comparison to a wavelength. One must then be concerned with the excitation and propagation of a number of waveguide modes in the beam tunnel. It is possible that this may limit the power gain in gyro- tron amplifiers to fairly low values (perhaps as low as 10 to 20 dB). 8-1 We have made design calculations for a gyrotron amplifier to produce 100 kW at 120 GHz based on the existing analytical models. Such a device appears to be achievable with reasonable beam current density and allowable dissipation power densities. Increases in power level up to 200 kW may be possible by increasing beam voltage and current and optimizing efficiency. Further increases in output power beyond that point will require increases in beam and circuit area. 8-9 10. REFERENCES S. Ramo and J. R. Whinnery, "Fields and Waves in Modern Radio, " pp 366-380, John Wiley and Sons Inc. (1953). F. Paschke, "Note on the Mechanism of the Multipactor Effect, " JAP, vol. 32, No. 4, pp 747-749, April 1961. A. Goldfinger, "High Power RF Window Study, " Final Report Contract No. AF 30 (602)-3790, Technical Report No. RADC-TR-66-657, Varian Associates, January 1967. P. K. Tien, "A Large Signal Theory of Traveling Wave Amplifier," BSTJ 35, 2, 349 (1956). S. E. Webber, "Large Signal Analysis of the Multicavity Klystron, " IRE Transactions on Electron Devices ED-5, 4, 300 (195S). J. E. Rowe, "Non-Linear Electron-Wave Interaction Phenomena, " Academic Press (1965). J, W. Sedin, R. E. Vehn, G. Wada, "Research on Interaction Circuits for Traveling Wave Tubes," Final Technical Report Contract No. AF(616)-6431, Watkins-Johnson Co., (1960). M. Chodorow, E. L. Ginzton, I. R. Nielsen and S. Sonkin, "Design and Performance of High Power Pulsed Klystrons, Proc. IRE, 41, 11, 1584(1953). J. Feinstein and R. J. Collier, "A Class of Waveguide-Coupled Slow- Wave Structures," IEEE Transactions on Electron Devices, -ED-S, 1, -9 (1959). A. V. Gaponov, M. I. Petelin, V. K. Yulpatov, "The Induced Radiation of Excited Classical Oscillators and its Use in High Frequency Electronics, Investiya VUZ Radiofizika, vol. 10, No. 9-10, pp 1414-1453, 1967. B. Kulke and C. M. Veronda, "Millimeter Wave Generation with Electron Beam Devices, " Microwave Journal, vol. 10, No. 9, pp 45-53, September 1967. R. M. Phillips, "The Ubitron, A High Power Traveling Wave Tube Based on a Periodic Beam Interaction in Unloaded Waveguide, " IRE Trans. ED-7, No. 4, pp 231-241, October 1960. 10-1 13. C. E. Enderby and R. M. Phillips, "The Ubitron Amplifier — A High Power Millimeter-Wave TWT," Proc. IEEE, 53, No. 10, pp 1648, October 1965. 'i 14. R. B. Dyott and M. C. Davies, "Interaction Between an Electron Beam of Periodically Varying Diameter and EM Waves in a Cylindrical Guide, " Trans. ED, vol. ED-13, p. 374, March 1966. 15. R. B. Dyott, "A New Type of Fast-Wave Amplifier Using a Rippled Electron Beam, " Proc. 6th Int'l Conference on Microwave Generation and Amplification, Cambridge, pp 158-164, September 1966. 16. D. T. Swift-Hook and A. Reddish, "Cyclotron Resonance and the Generation of Millimeter Waves," Proc. Symposium on Millimeter Waves, vol. 9, Polytechnic Institute of Brooklyn, 1959, pp 261-273. rj 17. K. K. Chow and R. H. Pantell, "The Cyclotron Resonance Backward Wave Oscillator," Proc. IEEE, vol. 48, pp 1865-1870, November 1960. 18. J. L. Hirshfield and J. M. Wachtel, "Electron Cyclotron Maser, " Phys. Rev. Let., vol. 12, No. 19, pp 533-536, May 1964. 19. J. B. Bott, "A Powerful Source of Millimeter Wavelength Electro- magnetic Radiation," Physics Letters, vol. Id, No. 4, pp 293-294, February 1965. 20. J. P. Beasley, "An Electron Cyclotron Resonance Oscillator at Millimeter Wavelengths, " Proc. 6th Int'l Conference on Microwave and optical Generation and Amplification, Cambridge, pp 132-139, September 1966. 21; R. L. Schriever and C. C. Johnson, "A Rotating Beam Waveguide Oscillator," Proc. IEEE, vol. 54, No. 12, pp 2029-2030, December 1966. 22. B. Kulke and R. W. Wilmarth, "Small Signal and Saturation Character- istics of an X-Bancl Cyclotron-Resonance Oscillator," Proc. IEEE, vol 57, No. 2, pp 219-220, February 1969. 23. V. L. Granatstein, P. Sprangle, R. K. Parker, and M. Herndon, "An , Electron Synchrotron Maser Based on an Intense Relativistic Electron Beam, "JAP, vol. 45, No. 5, pp 2021-2028, May 1975. 24. V. L. Granatstein, P. Sprangle, M. Herndon, R. K. Parker and S. P. Schlesinger, "Microwave Amplification with an Intense Relativistic Electron Beam," JAP, vol. 46, No. 9, pp 3800-3805, September ,1975. 10-2 D 25. P. Sprangle and V. L. Granatstein, "Stimulated Cyclotron Resonance Scattering and Production of Powerful Submillimeter Radiation, " Appl. Phys. Lett., vol. 25, No. 7, pp 377-379, October 1974. 26. N. I. Zaytsev, T. B. Pankratova, M. I. Petelin, and V. A. Flyagin, "Millimeter and Submillimeter Waveband Gyrotrons," Radiotekhnika i Elektronika, vol. 19, No. 5, pp 1056-1060, 1974. 27. D. V. Kisel', G. S. Korablev, V. G. Navel'yev, M. I. Petelin, and Sh.E. Tsimring, "An Experimental Study of a Gyrotron, Operating at the Second Harmonic of the Cyclotron Frequency, with Optimized Distribution of the High Frequency Field," Radio Engineering and Electronic Physics, vol. 19, No. 4, pp 95-100 (1974). 28. H. R. Jory, "Investigation of Electronic Interaction with Optical Resonators for Microwave Generation and Amplification," Final Report, Contract DA-28-043 AMC-01873(E), July 1968. '' 29. Marcuvitz, "Waveguide Handbook," McGraw-Hill, pp 66-72 (1951). 30. Ben Laboratories Staff, "Radar Systems and Components, " D. Van .Npstrand, pp 909-1020, 1949. 31. E. G. Avdoshin, L. V. Nikolaev, I. N. Platonov and Sh. E. Tsimring, "Experimental Investigation of the Velocity Spread in Helical Electron Beams, " Radio Physics and Quantum Electronics, vol. 16, No. 4, pp 461-466, April 1973. 32. A. L. Gol'denberg and M. I. Petelin, "The Formation of Helical Electron Beams in an Adiabatic Gun, " Radio Physics and Quantum Electronics, vol. 16, No. 1, pp 106-111, January 1973. 33. Sh. E. Tsimring, "On the Spread in Velocities in Helical Electron Beams, " Radio Physics, and Quantum Electronics, vol. 15, No. 8, pp 952-961, August 1972. 34. A. H. Beck, C. J. Edgcombe and N. D. Kenyon, "A Microwave Device Incorporating Adiabatic Compression of a Rotating Annular Beam, " Proc. 6th Int'l Conference on Microwave and Optical Generation and Amplification, Cambridge, pp 191-197, September 1966. 35. J. R. Pierce, "Theory and Design of Electron Beams," D. Van. Nostrandj 1954, pp 35-37. 10-3 36. P. F.,C. Burke, "Compensated Reversed Field Focusing of Electron Beams," Proc. IEEE 51, 11, p. 1653 (1963). 37. M. Muller, "Theory of a Classical Maser, " Central Research Memo- randum CRM-16S, Varian Associates J.1965) (see Appendix B of this report). i 38. V. I. Gayduk, R. F. Matveyev, A. T. Fialkovskiy, V. V. Dementyenko, "Irreversibility of Change of the Average Energy of an Ensemble of Non- linear, Mutually Noninteracting Oscillators Placed in a Radiation Field," Radiotekhnika i Electronika, 18, 4, 749 (1973). 39. V. A. Zhurakovskiy, "Using an Averaging Method to Integrate Relativistic Nonlinear Equations for Phase-Synchronous Instruments, " Radiotekhnika i Electronika, 9, 8, 1527 (1964:). 40. G. N. Rapoport, A. K. • Nemak, V. A. Zhurakovskiy, "Interaction Between Helical Electron Beams and Strong Electromagnetic Cavity Fields at Cyclotron Frequency Harmonics," Radiotekhnika i Electronika, 12, 4, 587 (1967). 41. E. M. Demiclovich, I. L. Kovalev, A. A. Kurayev, F. C. Shevchenko, "Efficiency Optimatizcd Cascaded Circuits Utilizing the Cyclotron Resonance," Radiotekhnika i Electronika 18, 10, 2097 (1973). 42. A. A. Kurayev, F. G. Shevchenko, V. P. Shostakovich, "Efficiency- Optimized Output Cavity Profiles that Provide a Higher Margin of Gyroklystron Stability," Radiotekhnika i Electronika 19, 5,^046 (1974). 43. V. L. Bratman, M. A. Molseev, M. I. Petelin, R. E. Erm, "Theory of Gyrotrons with a Nonfixed Structure of the high Frequency Field, " . Izvestiya VUZ Radiofizika, 16, 4, 622 (1973). 44. G. S.- Nusinovich and R. E. Erm, Elektronnaya Tekhnika, Sec. 1, Elektronika SVCti, 1972, No. 2, 55. 45. E. Feenberg, "Notes on Velocity Modulation, " Report No. 5221-1043, " Sperry Gyroscope Co. (1945) 46. I. S. Kovalev, S. V. Kolosov, A. A. Kurayev, "Calculation of the Transverse Electrical Fields of a Space Charge in Axisymmetric Gyroresonant Devices with an Annular Electron Beam, " Radiotekhnika i Electronika, 18, 7, 1525 (1973). 47. D. L. Webster, "Cathode-Ray Bunching," JAPf 10, 7, 501 (1939). 8-4 48. D. R. Hamilton, J. K. Knipp and J. B. H. Kuper, "Klystrons and Micro- wave Triodes," (vol. 7, MITRad. Lab. Series) pp 202-209, McGraw-Hill • (1948). v> 49. T. Wessel-Berg, "A General Theory oC Klystrons with Arbitrary, Extended Interaction Fields," Microwave Laboratory Report No. 376, Stanford University (1957). ' 50. T. Wessel-Berg, "Space-Charge Wave Theory of Interaction Gaps and Multi-Cavity Klystrons with Extended Fields," NDRE Report No. 32, p. 67. 51. H. R. Jory and A. W. Trivelpiece, "Charged Particle Motion in Large Amplitude Electromagnetic, Fields, " JAP, 39 No. 7, pp 3053-3060, June 1968. '<>• 8-5 APPENDIX A LIMIT ON POWER TRANSMISSION THROUGH CERAMIC SEALS Figure 1 shows a ceramic plate in which heat is flowing in the x direction. Due to the thermal expansion of the ceramic, stresses will be set up, and if the bending of the plate is neglected, it is simple to estimate these stresses. y Figure 1. Ceramic Plate Let e^ and e^ represent the strains in the y and z directions; jx, Poisson' s ratio; k^, the thermal coefficient of expansion; and AT, the difference between the temperature of the thin layer of ceramic in which the stresses are being investigated and the average temperature of the plate. Then (1 + e -n ez) (1 + k1 AT) = 1, (1) A-1 n BLANK PAGE o o o O ^ V;; o o o and (1 + e - n e ) \l + k AT) = 1. (2) v z ^ v 1 Since e = e =.fv (3) y z M where s is the stress in the layer and M is Young's modulus, either Equation (1) '.V or Equation (2) reduces to at =_ J— L ri , - „S/M (I - U) Ii (4) ~ k L 1 + s/M („1 - M) -J ' Since s/M (1 - jj.) < < 1, Equation (4) becomes AT = - ^nr^1 - * " (5) From this equation it is possible to calculate the temperature difference which corresponds to the ultimate strength of the ceramic. The value of n is assumed to be 0.25, the theoretical for isotropic materials. It can be-shown that if heat is generated uniformly throughout the volume of a ceramic plate and flows to one of the surfaces, the temperature distribution is parabolic, and the difference between the temperature of the surface through which the heat is leaving and the average temperature is given by P 2 tl ^3 <> where P = power loss in'"the ceramic in W/cm L u t = the thickness of the ceramic in cm k9 = the thermal conductivity of the ceramic in cal/sec-cm-°C ''3 The power loss in W/cm for a dielectric in an electric field is given by " PL-»4" •" * W"U I ~ (7> A-2 where GJ = angular^frequencVy in rad/sec • E = electric field in peak v/cm, and bi e" = loss factor If, we combine Equations (5), (6) and (7), we obtain 2 F _ 12.6 (1 - fx) /fmax_S_\ .2 Ik e" M ] wet \ 1 / o , in which e is the permitivity of free space in Farads/cm. » V—'' % G £ i/ a A-3 APPENDIX B J' " ( fX THEORY OF A CLASSICAL MASER* by M. W. Muller Varian Associates Palo Alto, California *Work supported by the U.S. Air Force under Contract AF 33(615)-2336 BLANK PAGE J ABSTRACT It has been pointed out by Lamb that an assembly of excited anharmonic oscillators is capable of providing coherent stimulated emission. A classical small-signal theory for the behavior of such a medium is given. The theory confirms the quantum mechanical result obtained previously for the cyclotron motion of relativistic electrons. It is applicable to any system of oscillators for which the energy dependence of the frequency is known. I. INTRODUCTION The word "maser" is an acronym for "microwave amplification by the stimulated emission of radiation". If this is taken literally, this word (and its extensions "laser", "iraser" etc.) describes all coherent amplification processes. It has been generally agreed to reserve the term for devices that contain a resonator and an "active medium". The active medium is prepared in such a way that when it interacts with radiation of some particular frequency but in any phase, stimulated emission dominates over absorp- tion. The medium ordinarily is a collection of independent or weakly coupled identical atoms (or molecules, ions, spins etc.). The preparation ("pumping") consists of ener- gizing the medium so that of a pair of energy states linked by a radiative transition, the upper state is more heavily occupied than the lower. After "pumping" but before the start of the radiative interaction, each atomic system is described by an incoherent superposition of states, that is to say, by a density matrix with no off-diagonal elements. The active medium, at the start of the interaction, has no oscillating electromagnetic moments; the moments (off-diagonal matrix elements) build up, in the proper phase for stimulated emission, as a result of the radiative interaction. The atomic systems that are ordinarily used in masers have only a small number of relevant energy states, and operate (apart possibly from some near-degeneracies) on t a single radiative transition between a particular pair of states. Under these conditions the usual quantum mechanical description in terms of "population inversions" is both simple and convenient. It is possible, however, to have long series of energy levels with "inverted" populations, and for many transitions to be involved at once in the radiative interaction. One such system, based on anharmonic oscillators, has been described by Lamb1 who pointed out that a classical theory is both logically and math- ematically more appropriate in this instance. Lamb also demonstrated, by means of a numerical calculation, that coherent stimulated emission can be expected to occur in a suitably prepared medium of this kind. B-l This classical model is of considerably more than, academic interest since it is applicable, among other things, to cyclotron motion of electrons in solids which are potentially an important source of submillimeter and far infrared radiation. The present paper gives a theory for this kind of maser. II. THEORY OF THE ANHARMONIC OSCILLATOR MASER 1. Description Consider a collection of identical anharmonic oscillators, each oscillator moving with the same energy, but with phases distributed at random1. The natural oscillation frequency of each oscillator is a function of its energy, increasing with energy if the oscillators are "hard spring", decreasing if they are "soft spring" type. Assume, for definiteness, that they are the latter. Now let this assembly of oscillators interact with a forcing field which oscillates at a frequency near the natural frequency of the oscillators. The oscillators will gain or lose energy, according to their initial phase. An oscillator that absorbs energy now has a somewhat lower natural frequency and conversely an oscillator that loses energy acquires a raised natural frequency. But if the driving frequency is higher than the initial frequency of the oscillators, those © which gain energy will lose their phase more rapidly than those which emit. Thus we expect the emissive interaction to be somewhat stronger than the absorption, and on the average the oscillators should transfer energy to the driving field. ' ' „ 2. The Correspondence Principle The classical finite-amplitude motion of a harmonic oscillator with well- defined phase can be described quantum mechanically by a coherent superposition of states in which th^'rojoability amplitudes are given by a Poisson distribution2 . The Poisson distribution peaks at the classical value of the energy. A collection of iden- tical oscillators with the same energy and with random phases has no oscillating dipole moment and may, for interactions with an external field, be represented by B-2 an incoherent superposition of states with the same probability densities (that is to say, by destroying the phases of the probability amplitudes). Population inversions occur for al] pairs of states up to the peak of the Poissdn distribution, non-inverted popula- tions above. Such a medium is always absorptive, because of the precisely equal spacing of the energy levels and because of the increase of the transition matrix elements with energy. If the oscillators are anharmonic, the spacing of the energy levels becomes non-uniform. For "soft spring" oscillators (e.g. Morse potential, simple pendulum) the spacing decreases with energy, for "hard spring" oscillatoi-s (e.g. quartic potential) it increases with energy. The classical motion with finite amplitude is again described by a distribution of probability amplitudes that peaks near "the classical energy, with inverted populations below this energy and non-inverted populations above. Thus stimulated emission can dominate at frequencies corresponding to transitions between low-lying energy levels, and absorption dominates at the upper levels. Note that this confirms the qualitative classical argument given above: The spacing of the low-lying levels of a soft-spring oscillator corresponds to the high-frequency side of the initial natural frequency. Calculation of the net energy exchange requires knowledge of the precise location of all energy levels and matrix elements. Such a calculation may be quite tedious even for the few potentials where these are known. The equivalent classical calculation given below is relatively simple. „ 3. Calculation of Energy Exchange The equation of motion of a driven oscillator is 2 A x + a x = — cos (vt + 6) (1) m ^ * f B-3 The rate of change of the energy of the oscillator is given by the rate at which the field does work on it dE • , , — = x A cos (vt + We shall assume that the initial conditions on all the oscillators are: t = 0, x = 0, x = v ; this permits us conveniently to replace the averaging over random initial O V; phases of the oscillators by an average'over $. If the oscillators are harmonic, Eq. (1) can be solved for the velocity. For our purposes a convenient way of writing the solution, is, using A = p, - v Av x = (1 - cos At) sin (Ft + ip) - sin At cos (vt + A + ——r cos sin (vt + At) (3) m(ji + v) r \ o If this is introduced into Eq. (2) we obtain dE I A v A sin At + — cos sin (At - d>) dt 2m A(jx + v) v A - (1 - cosAt) sin 2(vt + ~ v^A cos (At - + ~ k A cos (2ut + If A « v we can neglect the rapidly varying terms. W-ith this approximation we obtain for the slow variation of the energy I A v A 2 \ A E = E 1 + — cos cpl (1 - cos At) - ~sin 6 sin At o 2m 2 A (j.i + v) V A r 0 sin o (1 - cos At) + cos © sin At (5) B-4 and, after averaging over 6 2 E = E -r - (1 - cos At) (6) ° 4mA" which is a well-known result. o' To apply a similar procedure to the anharmonic oscillator we write j.i = v + A + f E(t) (7) where, of course, E(t) = E(t, t ^ A by A + f, At by At + J fdr. Let E = Eq + Ex(t) + E2(t) (8) where E^(t) is the time varying part of the harmonic oscillator energy - and E2(t) is the additional energy of the anharmonic oscillator. At this point, since we are inter- 0 ested in the buildup of oscillations from low levels, we will introduce a small-signal 2 approximation which consists of calculating E2(t) to order A and neglecting terms of higher order in A. The equation for the energy becomes dE 1 A2V . / = - IdTt 7T2m T(TA + f) (jjl + ^v) sm A\ t + J/ fcMJ 1 \2 1 + — 7-1 : cos d> sinlA/ t + ffd r - ©\ 2m (n + v) • y J J t + | VQA cos (At + /fdr- B-5 If A is small enough so that the change in energy is small, we can represent f(E) by a linear approximation m = a[E-Eoj=a (E1 + E£) (10) where of course a = = «(Eo)- Note that E has terms of order A and A so that.the lowest order terms in 2 E are of order A ; thus to obtain these terms, we can neglect E and the terms in E 2 2 of order A on the right hand side of Eq. (9), and to order A dE 2 = —1 v AA cos^At-f f E^(r)dr - cbj - cos(At -(b) dt 2 o a i r (11) Substitution for E^(t) from Eq. (5) yields, after some algebra dE2 aEoA sin At - At cos At dt 2 4mA + cos26(sin 'At + At cos At rv sin 2 At) + sin2o(At sin At - cos At + cos 2 At) (12) 1 2 where we have used E = — mv . O o 2 o Eq. (12) is readily integrated. We give only the result after averaging over 1 -J E"„9 aE A (1 - eos^ALj.^ At sin At (13) 2mA ° V " B-6 -1 2 4 The leading term of the expansion for small A is - (48m) ^ The complete expression for the average energy is 2 _2 E = Eq + ^ A A (l - cos At) - A^aE/l - "cos At - | At sin At] (14) A more useful expression is obtained by averaging E over "lifetimes" of the oscillators, with average lifetime r E = ^ J dt exp(-t/r) E(t) 9 9 1 A T~ 2 2 = E + ; r 1 + A r - 2a E rAT (15) o 0 4mL,v or 4 E m 1+x -2cx "Eo> (16)- .2 2 A T M where x = AT, C = AEor Eq. (16) has been previously obtained for the special case of the cyclotron motions of free electrons by Schneider3 , using energy levels and matrix elements obtained from the relativistic Schroduiper equation. The anharmonicity in.this case arises from the variation of mass with energy. Operation of a device based on this principle has been reported recently4,; It has been commented4 that in a free electron device with roughly constant transit time Eq. (13) may be more appropriate than the "Lorentzian" expression of Eq. (16). 0 B-7 The expression of Eq. (16), which is proportional to the average energy absorbed per oscillator, can become negative if 1) c and x have the same sign and 2) | cj > 1. The first of these conditions means that a an£l A must have the same sign. In terms of our previous discussion, a soft spring oscillator medium (a<0) can exhibit negative absorption for frequencies higher than its natural resonant frequency (A < 0 or M > /.t), and vice versa. The second condition specifies the minimum an- harmonicity, excitation energy, and lifetime for which negative absorption can occur at any frequency. The frequency at which the negative absorption peaks is readily obtained by differentiating Eq. (16). This leads to a cubic equation 3 2 x - 3cx + x + c = 0 (17) m m m whose smallest root of the same sign as c gives the requisite value of the frequency. -!/9 x ranges from 1 at c = 1 to 3 for c large. The corresponding minimum value m / Ki \ 2 2 of 4 ( E - E Jm/A r which starts'from zero at jc| = 1 is given quite accurately by 3/4 - 3 N/3~ JC j/S or .75 - .65 |c | for |c j > 1.5. To get some idea of the anharmonicity requirement, consider a potential / 2 4\ of the form 1/2 (x + yx I for an oscillator with m = 1 (small amplitude period 2-rr) : For small excitations of this oscillator, a = 3/2 y for a simple pendulum (^y - - "^j, ^ = " I * A model based on identical excitation of all the oscillators may be difficult to realize in practice. If there is a distribution of excitation energies, the absorption of energy must be averaged once more over this distribution. Let this distribution be f ^E^ , normalized so that f *?„*&)ml- B-8 Then the average energy absorbed is + 2 E = 712 A ?/ dE ff/E \)7 2 (l * )-2cx (IS) 4 J o v o) . v 2 o • / m H where c = a(Eo) Eor(Eo) (Eo) A (EQ) = , (EQ) -„ and where we have put m undei- the integral sign because the mass may be energy dependent. This average can only be carried out if the explicit energy dependences of r, a and m are known. In order" to get some estimate of the effect of an energy distribution, we shall assume Q?(E) = ai 7(E) = r, m(E) = m. Then we can write A(E) =M(0) + aE - v = o + aE (19) and using the notation QEqT = e, Eq. (18) becomes At " o 1 + (07 + e) where f'(e) = > ii and where we have carried out a partial integration. Eq. (20) shows that negative absorption can occur only if f'(e) has some positive region; this ig, roughly equivalent to the requirement for a population inversion. For a thermal distribution f'(e) is everywhere negative. ° \» n B-9 0 If the distribution is gaussian and fairly narrow £(Ec) =vfrexpH'32(Eo-H) or' 12 2 ? ~ 2 ~ AaT) ~ (21) where n = a = H, then 2, . 2 4mE X^ - 77) (22) a 2 2 A T 1 + (6r + €)" where £ = /3 /ar If £ is lartge enough (small energy spread), this integral can be carried out approximately. \In -the notation of Eq. (16) we obtain 4x4 + 4xr] - x2) - jl - x2) 4mE _ 2 3 1 + X - 2X7? + — (23) v 2 2 " . A T 2 r 1 + x 1 + X where x = 'AT = |M(H) ~ > that is to say, frequencies are measured from the natural frequency appropriate to the center of the gaussian. In terms of the standard deviation S of the gaussiari distribution, given by s2=/§o-H)24c)dE o the coefficient of the correction term in Eq. (23) is '.V 0 2 2 2 r = iars> . (arm | - r? | (24) B-10 The meaning of this correction is somewhat complicated because from a , physical point of view it contains two contributions. The first of these is only indirectly ° ' 1 associated with the anharmonicity. It arises from the fact tlat as a result of the distri- bution in energy the natural frequencies of the oscillators are also spread over a range of frequencies. Thus at any one frequency the interaction is reduced because only a portion of the oscillators interact strongly. This same effect would also occur if the oscillators were harmonic but distributed in frequency. The presence of a in £ as shown in Eq. (24) is partly accounted for by this circumstance, since a. relates the distribution of frequencies with the energy distribution. The other effect is more directly related to the mechanism we have been discussing. It tends to shift the fre- quency at which the strongest emission occurs away from the "center" frequency, that is to say, toward higher Jxj . This can be roughly explained as follows: The high- energy oscillators lie beyond the peak of the distribution, where f'(e) < 0. The con- tribution from this portion of the distribution is absorptive; thus the frequency of maximum emission tends to shift away from the region in which the interaction with ° this portion of the energy distribution is strongest. •• 1/ \f a B-ll III. REFERENCES )) 1. Lamb, W. E. Jr., in Quantum Optics and Electronics, 1964 Les^Houches Lectures, C. De Witt, A. BlandinandC. Cohen-Tannoudji, eds., Gordon and Breach, New York, 1965. 2. See, for example, E. M. Henley and W. Thirring, Elementary Quantum Field Theory, McGraw-Hill, New York 1962, p. 15 ff. 3. Schneider, J. , Phys. Rev. Letters 2, 504, 1959. 4. Hirshfield, J. L. and J. M. YVachtel, Phys. Rev. Letters 12, 533, 1964. a B-12