The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of DONALD RALPH McDIARMID B.A.Sc., The U n i v e r s i t y o f B r i t i s h Columbia, 1960 M.A.Sc, The U n i v e r s i t y of B r i t i s h Columbia, 1961 TUESDAY, JANUARY 19, 1965, a t 4:00 P.M. I.N ROOM 208, MacLEOD BUILDING COMMITTEE IN CHARGE Chairman; «K. C* Mann . E. V. Bohn C. F r o e s e M. K h a r a d l y External Examiner; F. Noakes Ro Nodwell A. C. Soudack R. M„ Bevensee Lawrence R a d i a t i o n L a b o r a t o r y , Livermore, California A STUDY OF CONFLUENCE I N PERIODIC SLOW WAVE STRUCTURES ABSTRACT An a n a l y s i s ture is of the uneven-offset given for the purpose extent to which s l o t of dispersion curve. the critical examination ditional confluence An e x i s t i n g coupled to cavity i n c l u d e the accuracy of of evanescent, of determining modes in this of chain presented this of the also permits p r e d i c t i o n of the cylindrical con- and t h e n extended is examined The experimentally. A d i s c u s s i o n of zero-mode conditional confluence based theory is presented. Experimental zero-mode confluence upon t h i s confirmation of The p o s s i b i l i t y of fluence i n the loop-coupled cussion cavity is made. achieving conditional centipede structure chain is and t h e discussed. conreversed- The is based upon an a n a l y s i s by Bevensee, confluence tests are complement existing confluent is presented to disTwo the ones. Finally 3 a d i s c u s s i o n on t h e structures presented. a slot- h i g h e r normal modes. extension shape structure. analysis effect structhe affect The a n a l y s i s a previous is corrugated usefulness for accelerator of auto- applications GRADUATE STUDIES Field of Study; Electrical Engineering A p p l i e d E l e c t r o m a g n e t i c Theory Non-Linear Systems Electronic Instrumentation G„ B„ Walker A. C, Soudack. F» K„ Network Theory Bowers A. D„ Moore Servomechanisms E. V. Bohn Communication Theory ' A„ D» Moore Electron G„ B. Walker Related Dynamics Studies: T, H u l l Numerical A n a l y s i s I Elementary Quantum Mechanics W„ Differential C. A. Swanson Equations Opechowski A STUDY OF CONFLUENCE IN PERIODIC SLOW VAVE STRUCTURES by DONALD RALPH McDIARMID B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, I960 MiA.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Electrical Engineering Ve accept t h i s t h e s i s as conforming to the required standard Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1965 In the presenting r e q u i r e m e n t s f o r an this thesis i n partial advanced d e g r e e a t B r i t i s h Columbia, I agree that available for reference mission for extensive p u r p o s e s may be and the g r a n t e d by Library study. copying of the this thesis Head o f my I t i s understood cation for f i n a n c i a l gain w i t h o u t my written Department of tz^/if The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada Date flpvJ 2. 6 ^. ( ) h c of that or c o p y i n g or s h a l l not \HI per- scholarly Department £ mciiw e @v Columbia, agree that for permission. oJ-y-ico I University of s h a l l make i t f r e e l y - I further his representatives. of t h i s t h e s i s the fulfilment be by publi- allowed ABSTRACT An a n a l y s i s of a c l a s s of corrugated s t r u c t u r e s i s given f o r the purpose of determining the extent to which s l o t evanescent modes a f f e c t the shape of the d i s p e r s i o n curve. also permits a c r i t i c a l examination of c o n d i t i o n a l confluence i n t h i s The a n a l y s i s of a previous p r e d i c t i o n structure. An e x i s t i n g a n a l y s i s of the c y l i n d r i c a l s l o t - c o u p l e d c a v i t y c h a i n i s presented and then extended e f f e c t of higher normal modes. i s examined e x p e r i m e n t a l l y . to i n c l u d e the The accuracy of t h i s e x t e n s i o n A d i s c u s s i o n of zero—mode confluence based upon t h i s theory i s presented. conditional Experimental c o n f i r m a t i o n of zero-mode confluence i s made. The p o s s i b i l i t y of a c h i e v i n g c o n d i t i o n a l confluence i n the centipede s t r u c t u r e and the reversed—loop-coupled chain i s d i s c u s s e d . by Bevensee. cavity The d i s c u s s i o n i s based upon an a n a l y s i s Two confluence t e s t s are presented to complement the e x i s t i n g one.. F i n a l l y , a d i s c u s s i o n on the u s e f u l n e s s of auto—conf l u e n t s t r u c t u r e s f o r a c c e l e r a t o r a p p l i c a t i o n s i s presented. TABLE OF CONTENTS Page L i s t of I l l u s t r a t i o n s ................ ...... v Acknowledgements o e a » « 0 » « - * e ' * » » < > * * o * « o « f t » * v * » * » « o * » * » » vxxi 1o INTRODUCTION 0 1 2. ANALYSIS OF CORRUGATED WAVEGUIDES VITH APPLICATION TO OFFSET STRUCTURES . . ............... 3. 0 O Q 4 > * « » 6 « O d 0 O f t a » » 0 O » O « » » » » » 0 0 0 0 O O * O 14 2ol In~b3?odue"tion ooQ»e'»»»*oo»o**«o**o»»9*o«»*«*e 14 2.2 A n a l y s i s of the Uneven-Offset S t r u c t u r e .... 18 2.2.1 F i e l d s i n the Line .................. 18 2.2.2 F i e l d s i n the S l o t s ................. 19 2.2.3 Matching of F i e l d Components ........ 24 2.2.4 The D i s p e r s i o n R e l a t i o n 30 2.2.5 Measurement of D i s p e r s i o n Curves .... 2.2.6 Numerical and Experimental Results .. 3 5 38 THE CIRCUMFERENTIAL-SLOT-COUPLED CONFLUENT STRUCTURE 56 3.1 Introduction 56 3.2 A n a l y s i s of the Coupled C a v i t y Chain 57 3.2.1 General Theory 57 3.2.2 Determination of the S l o t F i e l d 3.2.3 The D i s p e r s i o n R e l a t i o n ..... 61 65 65 3.3 D e r i v a t i o n of a Set of I 's nm One-Mode P r e d i c t i o n of Confluence . ..., 3.4 Multi-Mode D i s p e r s i o n R e l a t i o n s 82 3.5 The E f f e c t of an A d d i t i o n a l N o n - C r i t i c a l l y Resonant Coupling Element on Confluence .... 89 Concluding Remarks 91 302.4 3.6 iii 76 Page 4. AN INVESTIGATION OP THE ACCURACY OF THE T¥0-MODE ( T M - T M ) DISPERSION RELATION .............. 0 1 0 5. 1 1 0 97 A CONFLUENCE CONDITION FOR THE CENTIPEDE AND RELATED STRUCTURES I l l 6. AUTO-CONFLUENCE 120 7. CONCLUSIONS 127 APPENDIX I 129 APPENDIX XX o a « * « « * « * « O 0 o o * 9 o e a * e 4 o o « e « » « « o « e s o » * » * * e 137 APPENDIX I I I 140 APPENDIX XV • • • • • • • o * * o « o o o o a o o o o 9 * o c o o « o « o « * « « o * * * o * 143 o « o « * e o o » * » t t o o o 0 6 a o o o » o o o a a o t t o o o o o o o a o % * 4 9 146 APPENDIX V APPENDIX VI e e o » « » e * « « 0 o o o « 9 » o < t « o o o o o « o « o * o a o Q « « * * f t * * APPENDIX V XX REFERENCES » * * « * « « & * » e a « * Q O Q O o o » « e o o « o s a o o o 9 a * * 9 * * * » 0 O » » « * » a « O O » O « O O t t O O © O O O © O O O O < * « O « O O « » « » » « » iv 149 131 1 33 LIST OF ILLUSTRATIONS Page 1. A T y p i c a l D i s p e r s i o n Curve f o r a P e r i o d i c S t r u c "tUTG . o o « « » o e « * « o « o « o a a » o o o o * e o o a o o a a o o o » « * « « * * « « * 2. Resonant Frequencies 3. Space Harmonics of a P e r i o d i c S t r u c t u r e 4. -it-Mode C a v i t i e s f o r the D i e l e c t r i c Structure ..... 5. O f f s e t and Non-Offset Corrugated S"tl*LlC"tU.r©S » a « « * o a o o of a 10-Section o © a a o « o 0 9 0 6. The Uneven-Offset Structure 7. The Uneven—Offset Structure 8. The Even-Offset 9. The Microwave Bench a * 0 8 » C a v i t y ...... .......... 4 6 8 10 Periodic © « a o a o a o » » » « > * > * > « > » 11 12 .........«•• C a v i t y with P e r t u r b i n g Apparatus • 14 36 ........«•• 36 10. Test f o r Confluence i n the Uneven-Offset S t r u c t u r e 41 11. D i s p e r s i o n Curve f o r 42 12. Amplitude R a t i o s f o r \^ = 0.500 43 13. A Confluent 44 14. Amplitude R a t i o s f o r the Confluent 15. D i s p e r s i o n Curve f o r 16. Amplitude R a t i o s f o r 1^ = 0.635 17a. The The 0 d/V D i s p e r s i o n Curve Structure ..... 0.614 45 46 47 D i s p e r s i o n Curve f o r the Even-Offset C£tVX~by 17b. 0.614 Test o e o 9 « * # * « a « « t t e o a « a « o a a a * « « a » « a a o « a a t t « « * * # « a > 48 D i s p e r s i o n Curve f o r the Uneven-Offset Test X • o « * 0 # * « * * e e o o o o o o a a o o o a a o * « e o « o a o o a « « * * « > * « > 4*9 18. C r o s s i n g D i s p e r s i o n Curves 54 19. C i r c u m f e r e n t i a l - S l o t - C o u p l e d C y l i n d r i c a l Slow Wave S t r u c t u r e ........ = .............».••••• 57 20. Transmission 62 21. C y l i n d r i c a l S t r u c t u r e with Rotated 22. A T y p i c a l D i s p e r s i o n Curve f o r a C i r c u m f e r e n t i a l Slot-Coupled C y l i n d r i c a l S t r u c t u r e 76 23. Pass and Stopband Regions 78 Line Representation v of the S l o t ..... S l o t s ......... 73 Page 24. Zero-Mode Confluence 80 25. D i s p e r s i o n Curve l ( A > 1, a/L = 3.12, Z /r) = 0.552) 84 Q 26. D i s p e r s i o n Curve 2 ( A = l 2jQ^~f~j — 0 » 5 5 2 ) « 27. » o o o « a i d 0 t a/L = 3.12, a « « « o o o « » o « 0 9 « s « * a * * « » * * D i s p e r s i o n Curve 3 ( A = 1, a/L = 1.0, Z /77 = 0.552) 86 0 28. D i s p e r s i o n Curve 4 ( A > 1, a/L = 1.0, ZQ/TJ = 0• 552)—Rotated S l o t s ................... 29. Loading Disc f o r a M u l t i p l e - S l o t S t r u c t u r e 30. D i s p e r s i o n Curves f o r A 2 = 1.5 and 1.0 with A = 1.0 ..... 1 31. 32. 33 Mocle Separation Section Cavity as a F u n c t i o n of A f o r a 20......«•••• D i s p e r s i o n Curves f o r a Lossy C y l i n d r i c a l Coupled S t r u c t u r e 87 90 92 93 Slot95 33. C o n s t r u c t i o n Drawing o f the C y l i n d r i c a l C a v i t y • 98 34a. Experimental and T h e o r e t i c a l D i s p e r s i o n Curves f o r a Slot-Coupled Slow Wave S t r u c t u r e ......... 101 34b. Experimental and T h e o r e t i c a l D i s p e r s i o n Curves f o r a Confluent Slot-Coupled 35. A P l o t of ( f t - l)/(fi 36. D i s p e r s i o n Curves f o r a Loop-Coupled C a v i t y Chain An Even-Offset S t r u c t u r e and I t s D i s p e r s i o n Curve 37. 38. 39. 2 2 - Slow Wave S t r u c t u r e (fl> /» ) ) 103 109 2 3 113 121 A Doubly P e r i o d i c Slot-Coupled C y l i n d r i c a l S t r u c t u r e w i t h T y p i c a l C a v i t y Passband D i s p e r s i o n Curves 126 E q u i v a l e n t C i r c u i t of the Slot-Coupled 147 vi System .. LIST OF TABLES Page 1. S l o t and Line Element Amplitudes ........... 51 2. Consequences 125 V-l. Floquet Amplitude Ratio of Screw Symmetry 150 vii ACKNOWLEDGEMENTS I wish to express my a p p r e c i a t i o n to Dr. G. B. Walker f o r his support study, and encouragement throughout the course of t h i s I a l s o wish to express my a p p r e c i a t i o n to Drs, M.M.Z. Kharadly and E.V. Bohn f o r reading the manuscript and making many h e l p f u l suggestions. colleagues* In a d d i t i o n , I would l i k e to thank my e s p e c i a l l y Dr, C R , James and Mr. F.A* Goud, f o r the b e n e f i t of many e n l i g h t e n i n g d i s c u s s i o n s and suggestions. I should l i k e to express g r a t e f u l acknowledgement to the N a t i o n a l Research Council f o r Studentships and awarded i n 1961, 1962 1963> and f o r a Research A s s i s t a n t s h i p during thte l a t t e r p a r t of 1964. 1. INTRODUCTION P e r i o d i c s t r u c t u r e s , as the name i n d i c a t e s , are c h a r a c t e r i z e d by the f a c t t h a t t h e i r geometry i s repeated i n at l e a s t one d i r e c t i o n . Examples of such systems are c r y s t a l l a t t i c e s , i n t e r d i g i t a l waveguide s t r u c t u r e s , d i e l e c t r i c and metal-loaded waveguides and c o u p l e d - c a v i t y systems. This work i s concerned with microwave s t r u c t u r e s t h a t are p e r i o d i c i n one d i r e c t i o n only. In the a n a l y s i s of wave propagation i n such i t i s d e s i r a b l e to introduce the concept of a mode. structures, To see how t h i s may be done, i t i s h e l p f u l to re-examine the mode concept i n the theory of uniform hollow waveguides. In s o l v i n g Maxwell's equations f o r these guides, one seeks a simple wave s o l u t i o n ; i . e . , one whose z and t dependence are of the form j(<ot - B(o»)z) In so doing, one reduces the wave equation to the Helmholtz equation whose eigenvalues determine S(tt). the propagation f a c t o r , Corresponding to each e i g e n f u n c t i o n i s , i n the absence of degeneracy, a d i s t i n c t simple wave f u n c t i o n . f u n c t i o n s are fundamental and form a complete These wave orthogonal set so that complicated wave f u n c t i o n s can be expressed i n terms of them. These wave f u n c t i o n s are c a l l e d modes and are c h a r a c t e r - i z e d by t h e i r propagation f a c t o r , 3. I t i s important to note that any two modes (nondegenerate) do not combine to form a t h i r d ; i . e . , - j 3 (<o)z E(r,0,z) = E ( r , 0 ) e 1 + -jM<o)z E (r,0) e 2 ^ E (r,0) e 3 T i f ^ ( t t ) ^ p (o>) 2 In the case of p e r i o d i c waveguides, a s o l u t i o n of the form F C x ^ y e ^ ^ ^ ~ cannot, i n g e n e r a l , s a t i s f y the boundary c o n d i t i o n s and hence the simple mode theory of uniform waveguides does not apply. waveguide hollow However, by analogy with the hollow case, modes i n p e r i o d i c s t r u c t u r e s can be c h a r a c t e r - i z e d by t h e i r propagation p r o p e r t i e s ; namely, t h a t " ... the wave f u n c t i o n i s m u l t i p l i e d by a given complex constant when we move down the s t r u c t u r e by one period.""'". be r e f e r r e d to as the F l o q u e t c r i t e r i o n . This c o n d i t i o n w i l l To be more s p e c i f i c , a mode of a p e r i o d i c s t r u c t u r e w i l l be d e f i n e d to be a wave f u n c t i o n which s a t i s f i e s both Maxwell's equations and the Floquet criterion. As i n uniform hollow waveguide theory, a general p e r i o d i c wave f u n c t i o n can be expressed the i n terms of the modes of structure. Thus f o r a propagating mode i n a l o s s l e s s p e r i o d i c s t r u c t u r e , we may w r i t e E (z + L) = e ^ / ^ E (z) z 1 where L i s the p e r i o d i c l e n g t h of the s t r u c t u r e . function F(z) = e ^ z E (z) z and note t h a t i t i s p e r i o d i c with p e r i o d L. Consider the F(z + L) = e ^ e z J ^ E (z + L) L z = e^ E (z) z ' z = P(z) Thus P ( z ) can be expanded i n a F o u r i e r F(z) = / series. e a rn — oo Hence E (z) = / z / 7 — m a e a e . m o O oo m The term corresponding to any given value of m i s c a l l e d a space harmonic. I t s phase f a c t o r i s 8 = XJJ + 2nm/L. Note, however, that i n d i v i d u a l space harmonics are not wave s o l u t i o n s f o r the s t r u c t u r e and cannot e x i s t independently. space harmonic The term i s used because the harmonics are d i s t i n g u i s h e d by t h e i r phase f a c t o r and not by t h e i r frequency. As i n uniform hollow waveguide theory, the mode i s a continuous and d i s t i n c t f u n c t i o n of frequency. propagates only i n f i n i t e frequency bands. However, i t The modes a r e , i n g e n e r a l , d i s t i n c t from one another i n the sense that two of them do not combine to form a t h i r d . E (r,0,z) = e z 1 In other words ^ f m ml ( r '^ ) e + 4 I t i s shown by S l a t e r that i m p l i c i t i n the Floquet c r i t e r i o n i s the f a c t that the d i s p e r s i o n curve ( l ^ L vstt)i s symmetric about ^ph = 0 and i s p e r i o d i c with a p e r i o d = 2n f o r symmetric p e r i o d i c s t r u c t u r e s . In short, these s t r u c t u r e s are e s s e n t i a l l y f i l t e r s whose d i s p e r s i o n curves are s i m i l a r to the f o l l o w i n g ! — i -2% , 1 —ix Figure 1 . A T y p i c a l D i s p e r s i o n , 71. , 3 L m ,2-n;. Curve f o r a P e r i o d i c Structure. 5 Note that the phase v e l o c i t y , w / P , i s given by L times m the slope of the l i n e from the o r i g i n to the p o i n t of i n t e r e s t on the d i s p e r s i o n curve. Further, note that the group v e l o c i t y , d»/dl// , i s given by L times the slope the p o i n t of i n t e r e s t . Inspection of Figure frequency a> , the space harmonics of correspond to p o i n t s of the d i s p e r s i o n curve at 1 shows t h a t , at a forward wave (v > 0) on the d i s p e r s i o n curve given by c r o s s e s . Each of these space harmonics has a d i s t i n c t phase v e l o c i t y but an i d e n t i c a l group v e l o c i t y . I t i s convenient i n the f i r s t pass- band to take as the fundamental harmonic the one f o r which P^L l i e s between -ITand TZ» An important property of p e r i o d i c s t r u c t u r e s i s that the phase v e l o c i t y of some harmonics of a "Floquet than the speed of l i g h t \ mode" are l e s s Thus one use of a p e r i o d i c structure i s to produce electromagnetic wave-electron beam i n t e r a c t i o n . For example, i n the l i n e a r e l e c t r o n a c c e l e r a t o r , a bunch of electrons i s i n j e c t e d i n t o the slow wave s t r u c t u r e phased such that i t t r a v e l s i n the a c c e l e r a t i n g f i e l d of the harmonic to which the e l e c t r o n s are to couple. Obviously, i f t h i s a c t i o n i s to occur, the e l e c t r o n v e l o c i t y must be very n e a r l y i d e n t i c a l to the phase v e l o c i t y of the space harmonic i n q u e s t i o n . There have evolved two types of a c c e l e r a t o r p e r i o d i c slow wave s t r u c t u r e s . employing One i s the t r a v e l l i n g wave type i n which a pulsed microwave power source feeds energy i n t o one end of the s t r u c t u r e and the wave i n t e r a c t s with the e l e c t r o n bunch as they both progress along the s t r u c t u r e . end At the other of the s t r u c t u r e , the remaining energy i s e i t h e r absorbed i n a matched load or i s f e d back e x t e r n a l l y to the i n p u t . The other 6 type i s the resonant c a v i t y a c c e l e r a t o r which c o n s i s t s of a s t r i n g of c a v i t i e s , each composed of n p e r i o d i c s e c t i o n s terminated at each end on a plane of e l e c t r i c a l symmetry. the c a v i t y standing wave c o n s i s t s of two In t h i s t r a v e l l i n g waves; one corresponding to the forward wave (v > 0 ) of the slow wave s t r u c t u r e and the other to the backward wave (v <0) g structure. case, of the Here the e l e c t r o n bunch u s u a l l y couples to the fundamental space harmonic of the forward wave. In both cases, of course, the e l e c t r o n s are not i n j e c t e d u n t i l there has been a proper b u i l d up of the f i e l d s i n the slow wave s t r u c t u r e . I t can be seen from the d i s p e r s i o n curve shown i n Pigure 2 that an n - s e c t i o n c a v i t y resonances (n p e r i o d i c lengths) has n + 1 a s s o c i a t e d with the passband of i n t e r e s t . <0 — l//L Tt/2 Figure 2. 71 Resonant Frequencies of a 10-Section C a v i t y In F i g u r e 2, the r e s u l t s f o r a 1 0 - s e c t i o n c a v i t y ( i . e . , a c a v i t y of l e n g t h 10L.) are shown. 10 For a lb L = nn resonance or l/>L = S2L The range 0 ^ n ^10 i n t e g e r values general. and 0 ^ n ^ 10 permits n to take one i t can be of eleven or n + 1 seen that t h i s r e s u l t i s true i n By f u r t h e r examining Pigure 2, i t can be seen that the mode s e p a r a t i o n , by which i s meant the d i f f e r e n c e i n frequency between the resonance of i n t e r e s t and maximum at \JJL = %/2 In any and i t s nearest minimum at ij^L = 0 or neighbour, i s K. resonant a c c e l e r a t o r with a given number of s e c t i o n s , the g r e a t e s t mode s e p a r a t i o n p o s s i b l e i s d e s i r e d at the operating p o i n t i n order modes. to avoid the e x c i t a t i o n of unwanted Although t h i s c r i t e r i o n i n d i c a t e s that an should be operated at reason f o r operating there at l ^ L = TX. accelerator i s another important TO understand t h i s , i t i s h e l p f u l to define a parameter c a l l e d shunt impedance; namely, 2 where IE I i s the amplitude of the space harmonic of the com- ponent of the e l e c t r i c f i e l d with which the e l e c t r o n s i n t e r a c t and ¥ i s the power l o s s per u n i t length of the Provided structure. the energy absorbed by the e l e c t r o n beam i s ignored, which i s j u s t i f i a b l e f o r low beam c u r r e n t s , shunt impedance i s seen to be a measure of how w e l l any power input couples to the e l e c t r o n s . given Now, s t r u c t u r e with a given i n general, the back- ward wave i n a resonant e l e c t r o n a c c e l e r a t o r does not couple to the e l e c t r o n s , but does c o n t r i b u t e e q u a l l y with the forward wave to the o v e r a l l power l o s s . Thus 8 2 Z (resonant) = I 2F s h = ^ I ¥ ( t r a v e l l i n g wave) This c o n d i t i o n i s c l e a r l y a serious shortcoming of the resonant 2 accelerator. F o r t u n a t e l y , some workers at the it-mode (l^L = TT) m i t i g a t e s t h i s , , -2% —TT Figure 3. 1 : have noted that operp/tj disadvantage. , , — \ p h , 2lT TT Space Harmonics of a P e r i o d i c S t r u c t u r e R e f e r r i n g to F i g u r e 3, those p o i n t s oh the d i s p e r s i o n curve denoted by dots correspond to space harmonics wave of the forward and those denoted by crosses correspond to space of the backward wave. fundamental Usually electrons are coupled to the forward space harmonic (marked z e r o ) . i n c r e a s e d to » Q harmonics (TT-mode), the phase v e l o c i t i e s I f <o i s of the fundamental forward harmonic and the -1 backward harmonic become e q u a l . At 9 t h i s p o i n t , provided the e l e c t r o n v e l o c i t y i s c o r r e c t , e l e c t r o n s simultaneously couple to the two harmonics. Thus f o r a g i v e n ¥, the value of |EI may be c o n s i d e r a b l y i n c r e a s e d depending on the amplitude of the new harmonic. (resonant) In other words, has been c o n s i d e r a b l y improved. I f i t were not f o r the mode s e p a r a t i o n problem, -jx-mode o p e r a t i o n would l i k e l y be the one to use i n resonant 3 a c c e l e r a t o r s and, i n f a c t , i t has been used . In a study by 4 Valker and West to be d i s c u s s e d below, i t was shown f o r a s p e c i f i c design of a d i e l e c t r i c - l o a d e d s t r u c t u r e that the 71-mode shunt impedance was c o n s i d e r a b l y b e t t e r than f o r the other resonant modes. However, i n g e n e r a l , mode s e p a r a t i o n remains 5 a shortcoming. In 1956, Dunn, Sabel the propagation c h a r a c t e r i s t i c s of a s t r u c t u r e c o n s i s t i n g of a l i n e a r chain of i d e n t i c a l , coupled and Thompson investigated c a v i t i e s f o r which the c o u p l i n g elements, loops i n t h i s case, were resonant frequency near one of the resonant at a f r e q u e n c i e s of the c a v i t i e s . T h e i r a n a l y s i s made use of an e q u i v a l e n t c i r c u i t and showed that when the resonant equal frequency of the c o u p l i n g element i s made to the p e r t i n e n t resonant stopband i s eliminated. frequency of the c a v i t i e s , a They i n d i c a t e d , but d i d not prove, that the slope of the d i s p e r s i o n curve a t IpL = % under t h i s c o n d i t i o n i s non-zero; i . e . , t h a t mode s e p a r a t i o n a t the %—mode is increased. 4 In 1957, Walker and West d i s p e r s i o n curve showed t h e o r e t i c a l l y that the of a c y l i n d r i c a l waveguide p e r i o d i c a l l y loaded with d i e l e c t r i c d i s c s can have a stopband e l i m i n a t e d i f the wave impedance of the d i e l e c t r i c and a i r regions are made equal a t the 10 it-mode frequency. confluence holes. this I t was e s t a b l i s h e d t h a t the mode s e p a r a t i o n a t the it-mode i n c r e a s e d because of the f i n i t e group v e l o c i t y i n the c o n f l u e n t case. This c o n d i t i o n of stopband e l i m i n a t i o n has been c a l l e d confluence by Walker and West. I t i s u s e f u l to consider the experimental by Walker and West f o r the h o l e d - d i s c an i n f i n i t e (one that of passbands i s p o s s i b l e i f the d i s c s have c e n t r a l was c o n s i d e r a b l y there They f u r t h e r showed experimentally s t r u c t u r e , there f o r each passband). must be a t planes (see F i g u r e 4 ) . correspond structure. procedure used In g e n e r a l , i n i s a standing wave a t the Tt-modes The nodes of these standing waves of symmetry of which there are two sets Hence we see that the f o l l o w i n g two c a v i t i e s to the two TC—modes: F i g u r e 4. n-Mode C a v i t i e s f o r the D i e l e c t r i c Structure I f i t i s p o s s i b l e , b y v a r y i n g one or more parameters of the system,to make the resonant f r e q u e n c i e s the confluence did of the two c a v i t i e s c o n d i t i o n w i l l have been s a t i s f i e d . Walker and West j u s t t h i s by v a r y i n g the t h i c k n e s s of the d i s c . turns out t h a t the E equal, I t furthermore c o n f i g u r a t i o n of one of the resonances i s 11 s u i t a b l e f o r a c c e l e r a t i o n and the other d e s i r e d c o n f i g u r a t i o n i n an n - s e c t i o n i s not,6. To get the c a v i t y , the r i g h t s e t of t e r m i n a l planes must be chosen. 7 Lewis structure. l a t e r used the above t e s t to study the corrugated He considered double-sided corrugated corrugated structure the e f f e c t of o f f s e t t i n g the o r d i n a r y structure (Figure 5a) to form the o f f s e t (Figure 5b). a a '<, s / s 's / s ; '/"')s'"'- < / S / T7777T7T •777777777771 TT7-.777-! \7777 777777777 7777777777 a 5. i (b) (a) Figure a O f f s e t and Non-Offset Corrugated P e r i o d i c Structures He noted t h a t the two resonant c a v i t i e s of the o f f s e t circuit (given by planes a-a and by plane b-b) are i d e n t i c a l * Hence he concluded t h a t the o f f s e t s t r u c t u r e i s a u t o m a t i c a l l y confluent. To be more s p e c i f i c , he c a l l e d the s t r u c t u r e of Figure 5b the e v e n - o f f s e t geometrical s t r u c t u r e because i t s bottom h a l f has the same shape as the top h a l f . He f u r t h e r considered the case i n which t h i s symmetry d i d not occur and named the r e s u l t a n t c i r c u i t the uneven-offset s t r u c t u r e . I t can be seen 12 a X 2 a a Pigure 6. here that the two The Uneven-Offset S t r u c t u r e 71—mode c a v i t i e s are not i d e n t i c a l and, g e n e r a l , w i l l not have the same resonant a one-mode approximation v a r y i n g l^t frequency. By using i n the s l o t s , Lewis showed that by the other parameters being f i x e d , be made to occur. in T h i s type of confluence confluence i s similar can to those found by Walker and West and by Dunn et. a l . i n that the two c a v i t i e s used i n the confluence and confluence t e s t are not i d e n t i c a l i s obtained by f i n d i n g which produce i d e n t i c a l resonant s t r u c t u r e s w i l l be s a i d to be the c o n d i t i o n or c o n d i t i o n s frequencies. conditionally Such c o n f l u e n t confluent. Structures such as the e v e n - o f f s e t c i r c u i t which have i d e n t i c a l it-mode c a v i t i e s w i l l be c a l l e d auto-confluent• I t has been noted t h a t mdde s e p a r a t i o n can be i n c r e a s e d at the Tt—mode of a d i e l e c t r i c loaded making i t c o n f l u e n t s slow wave s t r u c t u r e by Other s t r u c t u r e s may be more u s e f u l f o r resonant a c c e l e r a t o r a p p l i c a t i o n i f t h e i r mode s e p a r a t i o n a t the 7t~mode can be i n c r e a s e d . work i s t o strengthen previous Consequently, the purpose of t h i s t h e o r e t i c a l p r e d i c t i o n s of con- fluence and to determine the c o n d i t i o n s f o r confluence periodic structures. i n other 14 2. ANALYSIS OF CORRUGATED WAVEGUIDES WITH APPLICATION TO OFFSET STRUCTURES 2.1 Introduction This chapter has two purposes. the confluence p o s s i b i l i t i e s 7 thoroughly than Lewis One i s to i n v e s t i g a t e of uneven-offset s t r u c t u r e s more has and the other i s to determine the e f f e c t of some of the approximations corrugated s t r u c t u r e a n a l y s i s . that have been made i n Although only the uneven-offset s t r u c t u r e i s analyzed, the a n a l y s i s procedure applicable. i s generally The f o l l o w i n g diagram shows the s t r u c t u r e and d e f i n e s the dimensional Figure 7. parameters: The Uneven-Offset Structure 15 The s t r u c t u r e to be t r e a t e d i s the t r a n s m i s s i o n l i n e which i s represented by the above diagram i f i t extends x direction (_L to p a p e r ) . guide w i t h two The i n f i n i t e l y i n the a n a l y s i s procedure f o r a wave- opposite w a l l s corrugated i s the same, b u t the t r a n s m i s s i o n l i n e adequately i l l u s t r a t e s the problem and i s s l i g h t l y 7 e a s i e r to handle. Lewis has i n d i c a t e d t h a t the d i s p e r s i o n curve f o r the waveguide can be obtained from the t r a n s m i s s i o n l i n e d i s p e r s i o n curve w i t h a frequency t r a n s f o r m a t i o n . A d e r i v a t i o n of t h i s t r a n s f o r m a t i o n i s o u t l i n e d i n Appendix I. An exact s o l u t i o n f o r the f i e l d i n the s t r u c t u r e can be obtained with the f o l l o w i n g procedure! I t has been c a l l e d the g classical s o l u t i o n by Walkinshaw and B e l l m e t a l - d i s c - l o a d e d waveguide. r e g i o n s , the s l o t s the l i n e (b y ^ The i n a d i s c u s s i o n of the structure i s divided into b + l p and -b s» y ^ -(b + I 2 ) ) two a n ( i (-b ^: y <g; b ) , and i n each the f i e l d s are represented by i n f i n i t e sums. In the l i n e , each element of the series corresponds to a p a r t i c u l a r Floquet phase constant; whereas, i n the s l o t s , each element i s given by e i t h e r the s l o t TEM wave or one of the i n f i n i t y of s l o t TM waves. E and H Z At the gap mouths, the f i e l d components f o r the l i n e are matched to those X f o r the s l o t s by making use of the o r t h o g o n a l i t y p r o p e r t i e s of the s e r i e s elements. This procedure y i e l d s two sets of i n f i n i t e homogeneous equations whose unknowns are the a r b i t r a r y i n the i n f i n i t e sums. The two sets of equations form amplitudes the i n f i n i t e e q u i v a l e n t of n homogeneous equations i n n unknowns. For the existence of a n o n — t r i v i a l s o l u t i o n f o r the the determinant amplitudes, of t h e i r c o e f f i c i e n t s i n the homogeneous equations 9 must be zero . This determinantal equation gives the between the phase constant, lb , and the frequency, w, relation and i s 16 c a l l e d the d i s p e r s i o n r e l a t i o n . amplitudes The r e l a t i v e values of the can be found a f t e r a s o l u t i o n to the determinantal equation i s obtained. However, a number of workers ( f o r instance Grosjean"*"^ and V a n h u y s e ^ ) have found that s o l v i n g f o r the zeroes of an i n f i n i t e determinant i s quite d i f f i c u l t . In order to produce a d i s p e r s i o n r e l a t i o n which y i e l d s s o l u t i o n s without excessive use of a computer, Lewis that E z i s constant across the mouths of the gaps. out, to j u s t i f y t h i s approximation, 7 assumed He p o i n t e d that i n p r a c t i c a l circuits the only propagating mode i n the s l o t s i s the dominant TEM mode. As a r e s u l t , he expected the amplitudes of the higher order modes to be small compared with amplitude of the dominant mode. He also used a p o i n t r a t h e r than an i n t e g r a l match f o r the H x f i e l d component (the exact meaning of the e x p r e s s i o n " i n t e g r a l match of the H follow). component" w i l l be c l a r i f i e d i n the theory to The matching p o i n t was the center of the s l o t . The 12 experimental r e s u l t s of L i n e s , N i c o l l and Woodward , which i n d i c a t e d i s p e r s i o n curve accuracy of about 5% f o r a s i m i l a r s t r u c t u r e and a n a l y s i s , were r e f e r r e d to by Lewis i n f u r t h e r j u s t i f y i n g h i s approach. Even with t h i s s i m p l i f y i n g assumption, Lewis' analysis y i e l d e d a d i s p e r s i o n r e l a t i o n which, although i n c l o s e d form, contained f o u r i n f i n i t e s e r i e s whose elements c o n s i s t of products of t r i g o n o m e t r i c and h y p e r b o l i c f u n c t i o n s . were t r u n c a t e d and the zeroes were found on a d i g i t a l through the use of an i t e r a t i v e technique. ±2 These s e r i e s computer Lewis showed t h a t can be v a r i e d with the other dimensional parameters f i x e d to y i e l d a c o n f l u e n t d i s p e r s i o n curve (it-mode). There that E is i s one u n s a t i s f a c t o r y consequence of the constant across the gap; assumption namely, t h a t the two sets of d i s t i n c t gaps (of width g and r ) have the same f u n c t i o n a l 7 form of gap f i e l d . Lewis s t a t e s t h a t " i t seems probable t h a t t h i s c r i t e r i o n w i l l not be met ing questions a r i s e : confluence by nature,•»."• Thus the f o l l o w - Is the f a c t t h a t Lewis' a n a l y s i s p r e d i c t s ( c o n d i t i o n a l ) an i m p l i c i t consequence of the assump- t i o n of i d e n t i c a l form of gap f i e l d s ? Is t h i s gap f i e l d d i t i o n a necessary c r i t e r i o n f o r confluence? con- I f the answers to these questions are yes and i f , as seems probable, the above gap f i e l d c r i t e r i o n i s not produced set i n nature, the s t r u c t u r e cannot be made c o n f l u e r i t i . uneven—off- Lewis thought a minor p e r t u r b a t i o n w i l l be the only r e s u l t " of the violation of the gap f i e l d c r i t e r i o n and t h a t the "that probable geometric c o n s t r a i n t necessary f o r confluence w i l l , i n p r a c t i c e , i n v o l v e all the dimensional One parameters. purpose of t h i s chapter i s to remove the doubt which leads to these q u e s t i o n s . In order to end up with a c o m p u t a t i o n a l l y manageable d i s p e r s i o n r e l a t i o n , the w i l l be l i m i t e d to approximating However, since the r e l a t i v e not a p r i o r i the purpose. the second analysis the s l o t f i e l d with two modes. amplitudes of the two modes are equal, t h i s development should be s u f f i c i e n t f o r The TMQ2 r a t h e r than the TMQ^ mode i s used f o r s l o t f u n c t i o n because, at the 71—mode, the E field z 13 component i s symmetric about the s l o t center Two . c a v i t i e s were b u i l t to provide an experimental test of the accuracy of the d i s p e r s i o n r e l a t i o n d e r i v e d i n s e c t i o n The experimental procedure theoretical i s described l a t e r . 2.2 Comparison of the and experimental d i s p e r s i o n curves p r o v i d e s an 18 i n d i c a t i o n of the dependence of the shape of the d i s p e r s i o n curve on the lower evanescent s l o t modes. 2.2 A n a l y s i s of the Uneven-Offset S t r u c t u r e 2.2.1 F i e l d s i n the Line (-b< y <b) The TEM mode i n the s l o t s provides E and H field z x A TM wave i n the l i n e has at r components at the s l o t mouth. least E . E and H f i e l d z* y x compatible components and i s the one most r with the TEM s l o t mode. A d d i t i o n of evanescent TM s l o t modes permits matching of the E^. f i e l d component. d e s i r e d TM l i n e wave has only one magnetic f i e l d The component, namely H , and i s d e s c r i b e d by b \ — \ 2 = S E 6 where E z S and 2 (2.1) 2 z = B(y)eJ< t t t z = p 2 - k " ^ 2 k = ft)/c A general s o l u t i o n to t h i s equation i s E (y) = P s i n h S(b + y) + P 1 s i n h S(b - y) z But, from the Floquet c r i t e r i o n , expression f o r E z i t f o l l o w s t h a t the complete must be an i n f i n i t e s e r i e s whose elements have phase constants of the form K =^ 2mn; + 19 Thus P E (y,z) = z s i n h S (b + y) + m z (2.2) - CO E where -JP P' s i n h S (b-y) ( y , z , t ) = E (y,z)e jat and S = P m m 2 2 r Prom Maxwell's H S - k 2 equations, 6y V^ 2 iB ^ z = J% — s 6y E E and y 2 Hence P m cosh S (b + y) m ' x J P m 1 cosh S (b-y) -JP z m ' m J d r m m (2.3). and "P m E ( y , z ) = +j y m •jB z m Jr P cosh S ( b + y) - m m P^ cosh S ( b - y ) m m . oo The p e c u l i a r choice i s made i n order of independent that P s o l u t i o n s of equation (2.1) be dependent only on the behaviour m of the s t r u c t u r e a t y = b and s i m i l a r l y f o r P^ a consequence, the algebra at y = -b. As involved i n obtaining a dispersion relation i s simplified. 2.2.2 F i e l d s i n the S l o t s For the purposes of t h i s d i s c u s s i o n , the d i r e c t i o n of propagation i s that of the y-coordinate f a c t o r e^ *^ ~ ^ ^ i s used. 6 a x i s and hence the The TEM mode i s d e r i v e d i n the usual 2 0 manner from Since the s l o t s are terminated v i t h a short c i r c u i t , the boundary c o n d i t i o n f o r the upper s l o t s at; y = b + 1^ = L ^ : i s E z = 0 . Thus E Prom Maxwell's curl sin k ( x - y) L l - | ^ z ^§ equations, H which f o r the TEM = A z _1_ P V (\ jcojx jtt|l Oz = x E 1 Y v - OE Z ) 6y ' mode y i e l d s -i- H ^ k7) x dy l - j ^ - cos K(L A ± - y) I t i s d e s i r a b l e t h a t the corresponding expressions f o r the lower from s l o t s c o n t a i n the i n f o r m a t i o n t h a t these the upper s e t . I f the lower s l o t s are s l o t at z = p / 2 i s to compare with the upper s l o t at z = 0 , the a r b i t r a r y constant f o r the lower Thus i t can be amplitude s l o t must i n c l u d e the f a c t o r e J ^ P / ^ , seen t h a t the expressions f o r the lower E z = A 2 e-<>rv L„ = b + 1 2 where S i n k(L 2 x = 3 A, ^ 2 e-^J>/ 2 cos k ( L 2 s l o t are + y) and H offset + y) ^ 21 Hence the s l o t TEM modes are given by a) upper s l o t (-q/2 ^ z < q/2) E z = s i n k(L-^ - y) x = - J1 A H b) lower s l o t -^j- = A e - ^ ^ 2 0A £ o the (p+r)/2) r z 2 e-^P/ e^W^^ 2 (2.4) x 1 ((p- )/2^ z < E I n t r o d u c i n g the f a c t o r , cos k ( L - y) s i n k ( L + y) 2 2 i s useful . cos k ( L 0 + . y) ( 2 * 5 ) because i t r e s u l t s i n r a t i o A /A-^ being a r e a l number. 2 The second s l o t mode i s to be TM and, hence, i s characterized by where K = n 2 T + k 2 2 'n E = E (z) e ""* ~ 7 n y n and 3 y v because there i s no v a r i a t i o n i n the x d i r e c t i o n . following The r e l a t i o n s are produced by m a n i p u l a t i n g the Maxwell c u r l equations i n a manner s i m i l a r to that done by Ramo and 14 Vhinnery : £)H z £) E 2 k 6 H 2 b < E 2 - 7 ) 22 The E y field component i s to be anti-symmetric about the center of the s l o t so as to y i e l d the d e s i r e d symmetric E z field component. The TMQ^ mode i s u n s u i t a b l e , whereas the T M Q mode 2 meets t h i s requirement. I t i s convenient to describe the E field component of the upper s l o t s with s i n i ^ i z + q/2) rather than s i n K z. L e t n E - = -B s i n K (z + |)e y n 2 Now E = 0 a t z = + a/2. y ~ y Hence H nix K and f o r the T M Q y n n q 2 q mode 2 7 2 (2.8) = (^) - k 2 2 2 Equations (2.7) y i e l d 2a |/ f i E K The boundary c o n d i t i o n , E it B y - y cos K ( z + 0 SL) e 2 = 0 a t y = L-^,must be s a t i s f i e d , as z i s by 2 7 Be E = ^-g1 2 1 9 z Since B i s an a r b i t r a r y cos K ( z + |) s i n h 2 constant, 7 ( L - y) 2 i t can be r e - d e f i n e d so as to y i e l d not only a more concise expression parameter with the same u n i t s as E z = B 1 cos ^ 1 . f o r E but a l s o a z Thus l e t (z + |) s i n h - y) 2 3 (2.7) Equations a l s o y i e l d the relation +y v where the negative s i g n corresponds H = + The q -qy 2 second mode f o r the lower ' 2 » 1 Hence J s l o t i s g i v e n by expressions very s i m i l a r to those above and can be from them by n o t i n g t h a t E ' (z + f ) cosh y _ ( L , - y) i- cos ^ x to the f a c t o r e = 0 at y = -L obtained and by u s i n g the 0 following transformations: L l 2 L t K — 2 z where K' z - ?• 2 ( 2 . 9 ) (^) 2 k - 2 boundary c o n d i t i o n f o r E be changed to s i n h , E z - J ^^^ 2 = B» e - 2 r e q u i r e s that s i n h z (L^ + y) and that cosh changed to cosh ' / ^ ( f a c t o r has e 2 = 2 y^ = The new - — K L 2 +y)o C O S "" ~ ^ As b e f o r e , the a r b i t r a r y added to i t . 3 ^ P / 2 T^^l T^^l ^ e amplitude Thus _ | ( z r g i n h _ (Ezr)) s i n + } y 2 « ( L 2 + y ) 2 = B 2 e C O S 2 * ( z h ^ ^ + y ) 24 and jkB,e-^P/2 Hence the second mode i n the s l o t s a) upper s l o t E H b) = B 1 z (-q/2 ^ cos x - c = B z 2 q/2) (z + |) s i n h lower s l o t E z ^ i s given by o ^ s ( z + f> c o - y) s h - y> > > i ( ( p - r ) / 2 < j z ^ (p+r.)/2) e-^P/ 2 C os ^ L ( z - ( ^ ) ) s i n h ^'(L + ? y) (2.11) ikB H ^ _ — 2 _ = g-^P/ v X 2.2.3 2 c o s i j ( z _ (Err)) c o s h ^ . ( L ^ y) Matching of F i e l d Components The matching procedure c o n s i s t s of four p a r t s ; namely, the separate matching of the E and H the top and bottom gap mouths. field components at both X. Z It is sufficient to match at one gap on both top and bottom because of the p e r i o d i c i t y both the s t r u c t u r e and the f i e l d s . of In other words, matching at two s l o t s produces a match at a l l s l o t s . The f i r s t f i e l d component to be matched i s a r b i t r a r y w i l l be the E the line E z component at the top s l o t which i s matched to component at y = +b. z (2.10) y i e l d and Equations (2.2), (2.4), and 25 E z = P ra -3P z s i n h 2S b m m _ oo A sin k ^ x + B ± cos — (z + |) 7^ sinh 2 ^ z ^ 2 •^z sg.p where 1^ = - b -JP z • 2mn -J - — z m Now e Thus = e P m s i n h 2S b m m A , s i n k l , + B, cos — (z 1 1 1 q + 2 sinh y _ l '21 n _ s. < z < r a 0 The s e r i e s i n m i s a F o u r i e r s e r i e s which i s known to c o n s i s t of a complete set of orthogonal functions'*'''. orthogonality, As a r e s u l t of the the f o l l o w i n g e x p r e s s i o n may be w r i t t e n : p P r m s i n h 2S b m = A-^ s i n k l ^ dz _ a. 2 SL '2 + B ± sinh yI 2 1 j 2mrt' e P '* cos ^L(z + a),q - a. 2 ^ dz 2A-^ s i n k l - sm P m x 2B B q/2 1 m 26 s i n P q/2 sinh — -2 9 ^ P 2 m m m 2 K 2 m 2 or P m 2 sin P q/2 A, s i n k l , 1 1 p s i n h 2S h m m m = > K The problem of matching H itself. B, P s i n h ' Y 1 I'm '2 1 x ? 2 K - 1 (2.12) 2 2 m at the upper gap now presents In h i s a n a l y s i s , Lewis matched t h i s f i e l d component at a p o i n t , that p o i n t being the center of the gap. because the f u n c t i o n a l dependence of H In any event, i n the i n t e r v a l -j25* ^ s s z s ^ p - ^. i s unknown, the o r t h o g o n a l i t y of the e f u n c t i o n s cannot be u t i l i z e d . z ^ However, the two-mode expansion i n the s l o t may be considered a truncated F o u r i e r s e r i e s at the gap. The two f u n c t i o n s , a constant and cos — (z + % ) , <s q are orthogonal over the range - ^=szsg;^. i s considered the a r b i t r a r y f u n c t i o n . Thus the s e r i e s i n m Note that t h i s procedure i s v a l i d i f an i n f i n i t e number of TM modes are used i n the s l o t s since the TM mode H cos ^ x field components w i l l vary as (z + ^) which are orthogonal over the range - ^ ^ ^ S . Z ; and complete over - ^<^z-^^m In the present s i t u a t i o n , equations (2,3), A l— V? (2.4) and (2.10) y i e l d cos k l , 1 1 k v. m -oo for - 2 < z ^2 * P m k B -qy l 2 cos — <1 cosh 2S b - P' m m m (z + %) cosh ' X l , 2 2 -iP JH m z 1 27 O r t h o g o n a l i t y r e l a t i o n s give A A n COS - / Ir k kl. •1 " 1 -J3 z m P cosh 2S b - P' m m m m d r dz m - oo _ a 2 P 2k q. cosh 2S b - P' m m m s i n 3 q/2 m mm m or 2k q cos kl-^ 1 m P cosh 2S b - P' m m m S 3 nr m (2.13) sin. 3 q/2 m — OO Also a. .2 B^k cosh T^^l cos 2 ^ (z + §)dz 2 q - 2 B^ qk cosh ?Tz 3. -2 P cosh 2S b - P' m m m = k m m — oo -JPm z Jt e ~ cos — ( z + | ) d z q 2 a 2 P = -2k m cosh 2S b m P' m B s i n 3 a/2 *m m^r m m 2 m - oo Hence oo B n = + 4 y 3 ( P cosh 2S b - P' ) m m m m m 2 m s i n 3 q/2 m q cosh T^^i m m 2 m (2.14) There now remains the matching at the lower s l o t Equations (2.2)> E (2.5) and (2,11) give f o r E f F« s i n h 2S b e I .m m = z m A ( sinkl + B 0 2 2 2jL (z - ( ^ 0 ) cos ^7 r sinh %'l 2 t2 0 where 1 = L 2 2 - b Again a F o u r i e r expansion i s produced P' m m - : 3 s i n h 2S b e m ^ P z — 00 A 9 sinkl + B 2 2 2jt cos ^J(z. - (2=^)) sinh / / oj.ij.ii r 2 %*1 J- ^ ( ^ ) < z < ( ^ ) 0 -(B=£.)<z<(B^) The f o l l o w i n g development r e s u l t s from the o r t h o g o n a l i t y of the e -j2nm z p J p P r functions: f ; m s i n h 2S b m p+r 2 A 2 sink l 2 e"^P e / 2 2 dz + 2 2 29 /' -^2 B s i n h yn e 9 9 jB / e z cos ^ m (z - (^))dz r 2 A - e sin k l 2 dz 2 r 2 r 2 + B 2 JB z' yn sinh 2 cos r 2 2 , (z* + x)d! r_ 2 where ;* = z - Now e"^ 2 e 3 = eJ ™ 2 1 £ = (-l) Comparison of the expression m above with that which l e d to equation (2.12) shows t h a t 2(-l) m m sin B r/2 m p s i n h 2S b m A 2 sin k l 2 m K 2 ? ' Pm' (2.15) A l l that remains now at the lower s l o t . equations (2.3), i s the matching of the H Applying (2.5) and x field component the p r e v i o u s l y used procedure with (2»ll) y i e l d s 30 B k 0 A 2 0 cos k l - cos & 2 (z - C ^ ) ) cosh ^ ' 1 2 2 '2 P s-ke r 2 3 - P' cosh 2S b m m m df m m At -30m z t h i s p o i n t , the t r a n s f o r m a t i o n z' = z - ^ can be used to advantage. The e x p r e s s i o n above becomes B k r,' A„ cos k l2 — — — 0 cos ~— (z + ) P' cosh 2S b - P m m m = k / , \m (-1) e -10 z m JK oo Again comparison with e a r l i e r development matching) T^'"^ m m — cosh 2 (upper slot shows that the f o l l o w i n g r e s u l t s are v a l i d : <?° r A 2 2k ~ r cos k l . P' cosh 2S b - P , m m m (-!)'" s i n 8 r / 2 SB mm V m m (2.16) _ OO and oO 4 % i (-l) S m ' B, r cosh ^ 1 r nv (P» cosh 2S b - P ) m m nr S (K' m 2 1 m 2 ( m 2.2.4 2 - 8 ) m 2 sin 8 r "m (2.17) The D i s p e r s i o n R e l a t i o n The matching has produced A B s i x equations, four of the form = A ( P . P») n n nr m = B (P , P') n n nr nr v (2.18) 31 and two of the form P = P (A, , B, ) ra m 1' 1 v (2.19) P m = m P <V V Two p o s s i b i l i t i e s now present themselves: E i t h e r equations (2.18) can be s u b s t i t u t e d i n t o equations (2.19) to form a s e t of homogeneous equations i n P substituted into in A n and B . n infinite m and P m or equations (2.19) can be (2.18) to form a set of homogeneous equations However, the former s u b s t i t u t i o n y i e l d s an 1 J s e t of equations; whereas, the l a t t e r y i e l d s four equations i n four unknowns. The l a t t e r set i s much more manageable and w i l l now be determined. Looked at more c l o s e l y , the approach here i s to r e t a i n the infinite l i n e expansion but to truncate the s l o t expansion. One could a l s o do the reverse i n which case the procedure would be to d e r i v e the s e t of l i n e a r homogeneous equations i n the P's m and P m ! , s . This i s , i n f a c t , what Valkinshaw ' and B e l l Q d i d when they t r e a t e d the c y l i n d r i c a l m e t a l - d i s c - l o a d e d structure. I f the amplitudes of one s e r i e s decrease c o n s i d e r a b l y more r a p i d l y than the amplitudes of the other, then i t should be the one t r u n c a t e d . it For the s t r u c t u r e of i n t e r e s t i n t h i s study, i s f e l t that the amplitudes of the s l o t modes w i l l decrease the most r a p i d l y (some numerical examples w i l l be given l a t e r ) . S u b s t i t u t i o n of equations (2.12) and (2.15) i n t o equation (2.13) gives 32 0 = A, q p cos kl-^ 4k coth n - sin kl. 2S b s i n S _m Pm S m 2 H m + A, q/2 2 (-l) sin k l , s i n 8 q/2 s i n P r / 2 m m m 3 S sinh 2S b •m m m 2 m . oo \ oo coth 2S b s i i T S a/2 S (K - 3 ) m 2 m 2 m - B, r (-l) s i n 3 q/2 s i n p^r/2 m S ( K l - B ) s i n h 2S b m m m m 2 yn 2 2 sinh 2 9 m m 2 2 m v — oo (2.20) S u b s t i t u t i o n of equations (2.12) and (2.15) i n t o equation (2.16) gives O O 0 = A, (-l) sin k l . s i n 3 q/2 s i n P r / 2 m 3m Sm sinh 2Smb m m m r m oo + A, p r cos k l 4k ""coth 2Smb s i n 3mr / 2 2 2 r - sin kl, 2 m - B, sinh y i 2 ± (-l) > 1 m s i n p q/2 s i n 3 r / 2 m S (K - p ) sinh 2S b ra 2 m ' m 2 2 r oo coth 2S b s i n 3 r / 2 m m . J r + B, sinh % ' l 2 2 m - i (2.21) S (K' - p ) m 2 m ' 2 v 2 r S u b s t i t u t i o n of equations (2.12) and (2.15) i n t o equation (2.14) 33 gives oo coth 2S„b s i n 0 q/2 m m 0 = A-, sin k l . m m — oo -4 s i n 8 q,/2 s i n 0 r / 2 (-1) sin k l , 2 m m — • S (K - 8 ) s i n h 2S b m m 2 ra m 2 2 r - m pq cosh " Y l , B, / 2 \ + sinh y i 1 2 8 % 8 N coth 2S b s i n 8 q/2 m ± m OO (-l) 0 1 + B, sinh y 'i 2 m s i n 0 q/2 s i n 0 r/2 2 x 2 —'S (K - 0 ) ( K I - 0 ) s i n h 2S b m m 2 m 2 m 2 2 m 2 2 m m m (2.22) And f i n a l l y , s u b s t i t u t i o n of equations (2.12) and (2.15) i n t o equation (2.17) gives oo 0 = - A-, sin k l 1 (-l) s i n 0 q/2 s i n 0 r / 2 m m m rn /_ , S ( K ' - 0 ) s i n h 2S b rn ' m m 2 m 2 2 m 1 + A. m coth 2S b s i n 0 r/2 m S (K« - 0 ) m 2 m sin k l . 2 m 2 r — oo (-l) 0 m + B, sinh y l x > s i n 0 q/2 s i n 0 r/2 ^ 2 m jS m m< 2 K 2 "O ^ ' "O s i n h 2 8 - oo - B, pr cosh yn o 1 0 8% o 0 + sinh yn 2 2 \ > m - oo 0 2 coth 2S b s i n 0 r/2 2 rn m S n,< 2 K 2 (2.23) 34 To c o n s o l i d a t e , i t : is. perhaps worth n o t i n g t h a t the l a s t f o u r expressions are e s s e n t i a l l y 0 = CT^A, + 0 = CT A 21 CT A 12 + 0- A 1 22 + 2 + 2 0 1 3 B <J B 23 + 0" B 1 14 + 1 CT B 24 2 2 (2.24) 0 = 0 A + 0- A 2 + 0 B o = aA 2 + aB 3 1 41 ± 1 32 + C^ A 2 33 43 1 + cr B 2 + C^ B 2 34 ± 4 I t i s a w e l l known f a c t that f o r existence of a n o n - t r i v i a l solution f o r the unknowns i n t h i s s e t of homogeneous equations, 9 the determinant of t h e i r c o e f f i c i e n t s must be zero, . In other words det ( CT ) = 0 (2.25) XX £ III This equation i s the d i s p e r s i o n r e l a t i o n . f u n c t i o n s of • ^ , k The O ' s are , and the dimensional parameters. If \p and the dimensional parameters are f i x e d , then the determinant i s s a t i s f i e d by a number of d i s t i n c t values of k; one for' each passband* Once the d e s i r e d s o l u t i o n f o r k i s found, r a t i o s of the A's and B's may be determined from the f o l l o w i n g set of equations which r e s u l t from equations 2 °12 i f A " °11 = " °21= °22A7 °31= °32Af l B °13 A7 + + + A " (2.24): °24A7 °23T[ + B + °33T[ 2 °14 i f B B + °34l^ ( 2 ' 2 6 ) In the event that k becomes g r e a t e r than 3 changes must be made i n the C T s. 1 f o r some m, c e r t a i n These changes are g i v e n i n Appendix I I . 2.2.5 Measurement of D i s p e r s i o n Curves The two c a v i t i e s t e s t e d c o n s i s t e d of t e n s e c t i o n s of both an even and an uneven-offset s t r u c t u r e with the f o l l o w i n g dimensions: a) The uneven-offset cavity p = 1.27 cm 1 q = 0.635 cm 1^ = 0*614 cm r = 0.9525 cm = 0.635 cm a = width of s t r u c t u r e b = 0.4064 cm b) = 1.778 cm The e v e n - o f f s e t c a v i t y p = 1.27 cm q = 0.635 cm - r = 0.635 cm 1 = 0.635 cm l = 0.635 cm a = 1.778 cm b = 0.4064 cm Ten s e c t i o n s were used i n order that enough experimental dis- p e r s i o n curve p o i n t s could be measured to c l e a r l y demonstrate the behaviour of the curve near The \jjj> = i t . c o r r u g a t i o n s were cut out of b l o c k s of brass and were pinned and screwed to a base p l a t e (side w a l l ) . The other side w a l l was formed by another p l a t e which was screwed to the b l o c k s c o n t a i n i n g the c o r r u g a t i o n s . The end w a l l s were provided by p l a t e s attached to each end of the c a v i t y . Pigure 8 i s a photo- graph of the e v e n - o f f s e t c a v i t y and the gear system which was used to p e r t u r b i t . The dimensions to ensure a v a l i d comparison were machined to + 0.001 i n . of the experimental r e s u l t s with the 36 POWER SUPPLY X13 SAWTOOTH ATTEMvATot; WAVE METER OSCILLOSCOPE ATTENUATOR MATCHING SECTION CAVITY Pigure 9. The Microwave Bench DETi CTOR 37 theoretical results. Two sets of i n s e r t s were made which blocked o f f a d i s t a n c e p/2 at each end of the c a v i t i e s to permit both TI—mode f r e q u e n c i e s to be measured. Input and output coupling to the c a v i t y was through c o a x i a l l i n e s terminated with a small loop antenna. The f i e l d p a t t e r n s are such that not a l l resonances corresponding to the d i s p e r s i o n curves under i n v e s t i g a t i o n can be e x c i t e d from the convenient coupling p o s i t i o n s . Consequently, two sets of coupling apertures were made with p r o v i s i o n f o r b l o c k i n g o f f the unused s e t . The microwave system i s shown i n b l o c k diagram i n F i g u r e 9. 28 Jhe method of measurement used i s d i s c u s s e d by G i n z t o n . A swept frequency output was obtained from the k l y s t r o n by modulating the r e p e l l e r w i t h a sawtooth wave from the o s c i l l o s c o p e . The microwave s i g n a l was coupled i n t o the c a v i t y and any e x c i t e d resonance was detected at the output and d i s p l a y e d on the o s c i l loscope ( i n the form of a resonance c u r v e ) . The center frequency of the k l y s t r o n was slowly v a r i e d u n t i l a resonance was found. The wavemeter, which had a much higher Q than the c a v i t y , was then tuned u n t i l a d i p appeared on the top of the c a v i t y curve. resonance Once the frequency of the resonance was found,, the wave- meter was detuned and the c a v i t y was perturbed by dragging a small metal needle through the l e n g t h of the c a v i t y along the central axis. The number of p e r i o d i c frequency s h i f t s caused by t h i s p e r t u r b a t i o n corresponds to the number of % phase along the c a v i t y . shifts The metal needle was attached to a l e n g t h of f i s h i n g l i n e which passed through the c a v i t y and was moved by the gear system. For each resonance, a p l o t of frequency p e r t u r b - a t i o n v s . needle p o s i t i o n was made and from these p l o t s the number of 71 phase s h i f t s was determined. 38 The metal needle p e r t u r b s other f i e l d components than E z and care must be taken to determine which component i s dominately perturbed. In the two c a v i t i e s i n v e s t i g a t e d , the E field z component, which i s maximum at the end w a l l s , i s dominately perturbed at some f r e q u e n c i e s and the E f i e l d y component, which i s zero a t the end w a l l s , i s dominately perturbed at o t h e r s . In the former case, the change i n p e r t u r b a t i o n from zero to maximum as the needle enters the c a v i t y must be ignored as i t i s not due to a variation i n E but to an only p a r t i a l presence of the needle z i n the c a v i t y . 2.2.6 Numerical & Experimental R e s u l t s An i t e r a t i v e procedure based upon a l i n e a r approximation of the f u n c t i o n det ( O " ) was used to produce d i s p e r s i o n curves. n,m Numerical v a l u e s were s u b s t i t u t e d f o r l^/p» P, q> r , 1^, 1 , a n < 2 i n t o det (CT ) so that i t became a f u n c t i o n of k o n l y . l D In other words det )— (CT F(k) l l • III Near a zero of F ( k ) , say k^, F(k) can be expanded i n a T a y l o r series. F o r the purposes of the i t e r a t i v e zero f i n d i n g procedure, only the f i r s t two terms of the expansion are r e t a i n e d ; i . e . , F(k) = Q 1 where k-j^ = approximate C ± + C (k - k ) 2 value of k^ = F(k ) x and C 2 = F'O^) (2.27) 39 The s o l u t i o n procedure i s guessed. first i s as f o l l o w s : Using t h i s v a l u e , The value of the zero and are computed and the i t e r a t i v e s o l u t i o n f o r the zero i s computed w i t h the expression k = k i- c 2 which r e s u l t s from equation (2.27) when P(k) = 0. The new estimate of the zero i s used to c a l c u l a t e new values of C-^ and C 2 which are then used to determine zero. another new value of the The i t e r a t i o n s are c a r r i e d on u n t i l where e r r i s s u f f i c i e n t l y small to ensure acceptable range of the a c t u a l |F(k)| < err, that k i s w i t h i n an zero. The present problem i s complicated by the presence of infinite series i n det (0~ ). n,nv v In p r a c t i c e * the s e r i e s were t r u n c a t e d to the range -N<mssN where N i s s u f f i c i e n t l y large so as to make no s i g n i f i c a n t change i n the zero s o l u t i o n i f i t were larger* The i n f i n i t e series 5 make the d e t e r m i n a t i o n of C l a b o u r i o u s and i t was approximated L 2 - 2 very by P(k, + Ak) - F(k,) Ak The number of s i g n i f i c a n t f i g u r e s , WL, used i n the c a l c u l a t i o n a l s o had to be s u f f i c i e n t l y l a r g e to produce an acceptable computation error. Numerical t e s t s were made at l ^ p = % f o r p = ..1.27 m 1 ± , q = 0*635 m = 0.635 m , 1 = 0.500 m with v a r i o u s values of N, WL, and e r r . , r = 0.9525 m , b = 0.4064 m Since the mks system i s being used, the dimensional parameters are i n meters. However, 40 an i n s p e c t i o n of equations and i f the dimensions (2.20) - (2.23) shows that i f l^/p = 71 are i n cgs u n i t s , equation (2.24), k and the r a t i o s of the A's and B's are unchanged. be r e a l i z e d that k i n t h i s case i s i n cgs u n i t s . , i> •<•. But, i t must The homogeneous equations are unchanged because the e f f e c t of the above t r a n s formation ( i . e . p i n meters becomes p i n centimeters) i s to leave the C T ' s u n a l t e r e d except f o r a common m u l t i p l i c a t i v e constant which can be ignored i n the computations. Thus the numerical t e s t s are good f o r both mks and cgs dimensional parameters. The r e s u l t s of the t e s t i n d i c a t e d that N = 11, VL = 14 and e r r = 10 are s u f f i - c i e n t t o produce k with three s i g n i f i c a n t f i g u r e accuracy i n comparison with the k obtained i f the s e r i e s were not t r u n c a t e d . Other spot checks The i n d i c a t e d b e t t e r accuracy. above dimensional parameters were used (with v a r i a t i o n s i n 1^) to compute a number of d i s p e r s i o n curves which are shown i n F i g u r e s 10 -17 along with the amplitude f l u e n c e t e s t curve. r a t i o curves and a con- The r e s u l t s obtained from equation (2.24) are shown as s o l i d curves and the r e s u l t s obtained from Lewis' p e r s i o n r e l a t i o n are presented as dashed curves 17a & b ) . dis- (no. 3 i n F i g u r e s Note that f o r l ^ p ^ n ; , there i s a c o n s i d e r a b l e d i f f e r e n c e between the curves; a d i f f e r e n c e which does not seem with the v a l u e s of B^/A-^ and B / A 2 2 obtained. compatible This d i f f e r e n c e i s e x p l a i n e d by the f o l l o w i n g d i s c u s s i o n . One d i f f e r e n c e between the a n a l y s i s presented here and t h a t g i v e n by Lewis i s the method used to match the H x field component* Lewis used a p o i n t match whereas equation (2.24) r e s u l t s from an i n t e g r a l match. I f one uses an i n t e g r a l match and one s l o t mode, the f o l l o w i n g equations are obtaineds F i g u r e 10, Test f o r Confluence i n the Uneven-Offset Structure Figure 11.. D i s p e r s i o n Curve f o r l p < 0.614 43 44 Pigure 13. A Confluent D i s p e r s i o n Curve 45 46 Figure 15. D i s p e r s i o n Curve f o r 1 > 2 0.614 47 F i g u r e 17a The D i s p e r s i o n Curve f o r the Even-Offset Test C a v i t y 49 Pigure 17b The D i s p e r s i o n Curve f o r the Uneven-Offset Test C a v i t y 50 = 0 °~11 1 A °~12 2 + A (2.28) 0 = This equation was G-2lAl + C7 A 22 2 solved f o r the c a v i t y s t r u c t u r e s and the r e s u l t s appear as curve 2 i n F i g u r e s 17a and b. Now examine F i g u r e 17a. the e x p e r i m e n t a l l y determined The c i r c l e d dots on t h i s p l o t are p o i n t s and the s o l i d l i n e above curve 1 shows the upper l i m i t of the experimental curve. A line i s shown because a number of resonances were found which were too close together to permit a p e r t u r b a t i o n a n a l y s i s * The response curves were so close that they became e x c e s s i v e l y d i s t o r t e d when the c a v i t y was perturbed. Both ix-mode f r e q u e n c i e s were, w i t h i n experimental accuracy, i d e n t i c a l as expected* Curve 1 was obtained from equation (2.24), curve 2 from equation (2.28) and curve 3 from Lewis' d i s p e r s i o n r e l a t i o n . Comparison of curves 1 and 2 with curve 3 shows that f o r the e v e n - o f f s e t s t r u c t u r e the i n c r e a s e d accuracy of the theory presented here i s p r i m a r i l y due to a proper i n t e g r a l matching of the H field component. The reason f o r the X success of the i n t e g r a l matching match minimizes technique i s t h a t whereas Lewis' the d i f f e r e n c e between the H the i n t e g r a l match minimizes expansions at a p o i n t , the mean square difference"'"*'. In f u r t h e r c o n s i d e r i n g the accuracy of the present approach when a p p l i e d to the s t r u c t u r e s being i n v e s t i g a t e d , the r e s u l t s i n Table 1 are s i g n i f i c a n t . E (P expansions Q (P 1 corresponds to the P TMQT amplitude f a c t o r s which appear i n the at y = + b are sinh 2S b), Q The sin kl^), s i n h 2S b ) , e t c . 2 (B The T M n s i n h T^^i^» Q 2 mode i n the or P^ element i n the l i n e expansion mode having been i g n o r e d ) . slots (the s l o t The d i s p e r s i o n curve p o i n t used 51 f o r the computation i s one f o r which B^/A-^ and B^J value. At the upper s l o t , (P compares w i t h (B^ s i n h sinh 2S b/P 2 'Y^l^l 2 s ^" n ^1^) reach a peak s i n h 2S b) = -0.206 0 Q 0»015 and = at the lower slot, 1 Tt/p 1.28(10" ) B /A 3.86(10" ) P{/P 3 V i A 2 3 p2 /0p r B^ -4.35(10"" ) 2 sinh 0.0154 A-^ s i n k l ^ P sinh 2S h 1 P Q s i n h 2S b P 2 s i n h 2S b P Q s i n h 2S b 0.346 Q Q -0.206 2 Table 1. (P* s i n h 2 S b / P 2 (B 2 sinh 1 Q 0.614 1.5(10 ) -2 2 1.77(10~ ) 3 0 2.0(10" ) P-*/P* 5 / r - 2 5 J B 2 0 sinh A 2 sin k l 1 2 P| sinh P s i n h 2S b Q 2S b 1 0.158 0 P£ s i n h 2S b P s i n h 2S b Q 0.055 2 2 -0.095 Q S l o t and Line Element Amplitudes s i n h 2S b) = - 0.095 compares with 2^2 s ^ n k l 2^ 0-055. = i s not anywhere near c o n c l u s i v e , b e t t e r to truncate , Q the While the evidence presented i t does i n d i c a t e that i t i s s l o t expansion r a t h e r than the l i n e expansion to decrease e r r o r . Now how consider some of the other r e s u l t s . - Figure the 7t—mode (\jj-p = it) frequencies q = 0.635, r = 0.9525, 1-^ =0.635 and v a r y w i t h the 1^ f o r p = b = 0.4064. t h a t equation (2.24) p r e d i c t s confluence when 1 The 10 shows 2 I t can be = 0.614 1.27, seen and k ='11521* curve .shows, i n agreement with Lewis* a n a l y s i s , that one ji-mode frequency,is independent of 1 » 2 "By symmetry, i t can be the other iwnode frequency i s independent of 1,. seen that Thus i t i s seen t h a t the p r e v i o u s l y d i s c u s s e d c r i t i c i s m of Lewis 1 p r e d i c t i o n of confluence i n the uneven—offset s t r u c t u r e i s not v a l i d . The assumption of i d e n t i c a l gap f i e l d c o n f i g u r a t i o n s has not been made, y e t the a n a l y s i s s t i l l Because of the convergence p r e d i c t s confluence. properties'"''' of the F o u r i e r s e r i e s with which the s l o t modes are expressed, the a d d i t i o n of f u r t h e r modes s t a r t i n g with the T M Q ^ mode should/produce p r e d i c t i o n of 1 2 °^ "^ a,more/accurate confluent structure. ne F i g u r e s 11-16 show d i s p e r s i o n curves f o r 1 to, slot 2 l e s s than, equal and g r e a t e r than 0.614 and the corresponding curves f o r A /A^, B^/A^, and B / A » 2 2 analysis. The dashed curves r e s u l t from 2 Lewis' I t i s i n t e r e s t i n g to note t h a t F i g u r e s 11, 12, 15 and 16 show t h a t A /A-^ = 0 corresponds to the s t r a i g h t l i n e i n F i g u r e 2 10 and t h a t k^l^ ~ 0 0 C O R R E S P O N ^ S to the s l o p i n g curve. Again we see t h a t one ix-mode frequency i s dependent upon 1^ and the other upon l n . F i g u r e 17b presents the t h e o r e t i c a l and experimental for the uneven-offset s t r u c t u r e whose dimensions results are given on page 35. The dimensions are those f o r the p r e d i c t e d c o n f l u e n t structure (see F i g u r e 13). Again the c i r c l e d dots represent the e x p e r i m e n t a l l y determined points. Curve 1 was obtained, as b e f o r e , from equation (2.24), curve 2 from equation (2.28) and curve 3 from Lewis 1 dispersion relation. Comparison of curves 1 and 2 with curve 3 shows t h a t f o r t h i s uneven-offset s t r u c t u r e , the i n c r e a s e d accuracy of the theory presented here i s due p r i m a r i l y to a proper i n t e g r a l matching of the H X field component. I n c l u d i n g the TMQ 2 mode i n the theory leads to a f u r t h e r s l i g h t i n c r e a s e i n accuracy for most v a l u e s of ^ p , but at the expense of computational effort. A proper one s l o t mode a n a l y s i s has produced q u i t e accurate 53 d i s p e r s i o n curves i n the two I t might be thought cases examined e x p e r i m e n t a l l y . t h a t the r e s i d u a l d i f f e r e n c e between curve 1 and the e x p e r i m e n t a l l y determined s t r u c t i o n of the c a v i t y . of 135 Mc to i n a c c u r a t e However, the curves shown i n F i g u r e i n d i c a t e t h a t an e r r o r of 0.001 structure w i l l curve i s due in. i n 1^ of the l e a d to a stopband of 16 Mc, The con10 uneven-offset observed stopband i s too l a r g e to be explained by t h i s source of e r r o r . A more accurate d e t e r m i n a t i o n of the dimensions of a c o n f l u e n t s t r u c ture should be obtained by using more s l o t modes i n the theory and can be obtained by e x p e r i m e n t a l l y determining the, e f f e c t on the •jtrmode f r e q u e n c i e s of s l i g h t , changes i n 1^ of the .uneven-offset cavity. I t i s to be noted t h a t F i g u r e 13 shows t h a t the group v e l o c i t y at the c o n f l u e n t TC—mode i s f i n i t e but non-zero which means t h a t energy propagation o c c u r s . The above f i g u r e s do not show c r o s s i n g d i s p e r s i o n curves because only the curves f o r v have been drawn. > 0 However, i t i s easy to see from the f o l l o w i n g argument t h a t there i s , i n f a c t , c r o s s i n g curves at l^p = TC: s i d e r the c o n f l u e n t d i s p e r s i o n curve of F i g u r e 18(a). d i s p e r s i o n curve i s symmetric about "^p = 0 f Con- Since the F i g u r e 18(b) may drawn and s i n c e i t i s p e r i o d i c with p e r i o d l^p = 2TX, F i g u r e be 18(c) i s valid,, F i n a l l y , l e t us consider the TC—mode grOup v e l o c i t y i n more d e t a i l . At the it-mode, 3 ^ = -3Q r e s u l t , s i n P_ q/2 = - s i n 3Qq/2 and from equation J (-l) m f a c t o r i n equation P' , / P * = -1. —m—1 m , / P =1. -m-1' m (2.15), P _ | / P ^ J 1 = -1 and, 7 Consider the f i r s t 0 = + !• Because of the i n general, U s u a l l y , at the Tt-mode, e i t h e r the P m or the P ' s are z e r o . m f 1 (2.12), i t f o l l o w s t h a t P ^ / P Q S i m i l a r l y i t can be seen t h a t P as a s i n h 2SQb = s i n h 2 S _ b 1 Consequently and SQ = S ^ and, case* ! t s are zero Here the space r 54 (c) (b) -2n IJJV Figure 18. Crossing Dispersion Curves harmonics combine i n the f o l l o w i n g manner to form a standing with c o s i n u s o i d a l a x i a l dependence. (P_ P (e 1 + P 1 + e u Q = 2 P e Q From equation (2.2) e u ) s i n h S ( b + y) Q ) s i n h S ( b + y) v Q cos P z s i n h S ( b + y) Q Q Thus Q oo P E s i n h S ( b + y) cos 8 z m m m m = 0 On the other hand, i f the sinusoidal axial 5 s exist, the standing dependence 3 -1 B (P_{ e Z - ^ 0 ) s i n h S ( b - y) z + p e 0 wave has a wave 55 P ( i (_ + u e = -2j u e . ) s i n h S ( b - y) Q s i n h S ( b - y) s i n S z Q Thus Q oo E P^ = 2j z s i n h S ( b - y) s i n ^z m m = 0 Each of these standing frequencies waves occurs at one of the it-mode and, i n both cases, the group v e l o c i t y i s zero. Now consider the confluent s t r u c t u r e f o r which both the primed and non-primed P ' s e x i s t a t the u-mode; etc» = a i.e.,PQ PQ, Thus at the it-mode ( P _ + P e 1 + = P (e Q ( Q P e ^ ) s i n h S ( b + y) 0 e + e . + P 1 Q e U ) s i n h S ( b - y) Q ) s i n h S ( b + y) Q +3'0 z " O N J 8 o Z + a P Q = jQ z ( s i n h S ( b + y) -a s i n h S ( b - y ) ) e (-e + e u ) s i n h S ( b - y) u Q 0 P 0 0 Q + ( s i n h S ( b + y) +a s i n h 3 ( b - y ) ) e Q 0 Q and + E z = [sinh S ( b + y) - a m m s i n h S ( b - y)] J m B z m m = 0 + s i n h S (b + y) +a ffl m s i n h S ,(b - y ) ] Now the space harmonics do not combine wave. Since the h y p e r b o l i c r e s u l t of the above argument functions •30 z n to produce a pure standing of y are always p o s i t i v e , the i s that there i s a net energy flow. 56 3. THE CIRCUMFERENTIAL-SLOT-COUPLED CYLINDRICAL CONFLUENT STRUCTURE 3.1 Introduction C y l i n d r i c a l a l l - m e t a l s t r u c t u r e s have a number of p r o p e r t i e s which make them d e s i r a b l e f o r use i n a c c e l e r a t o r s . One such property i s the a b i l i t y , i n a number of cases, to support waves w i t h no transverse E f i e l d on the a x i s * r e s u l t , beam f o c u s i n g problems are minimized. As a Also the s u r f a c e - to-volume r a t i o f o r a c y l i n d r i c a l s t r u c t u r e i s lower than f o r an e q u i v a l e n t s t r u c t u r e of n o n - c i r c u l a r cross s e c t i o n and, as a consequence, one expects the c y l i n d r i c a l s t r u c t u r e to have the higher shunt impedance. and nuclear physics I t i s o f t e n d e s i r a b l e i n medical a p p l i c a t i o n s of a c c e l e r a t o r s to focus the emitted e l e c t r o n beam on a p o i n t . c i r c u l a r cross the In t h i s case, a beam of s e c t i o n , f o r which c y l i n d r i c a l s t r u c t u r e s have c o r r e c t symmetry,is very convenient. In view of these f a c t s , knowledge of the confluence p o s s i b i l i t i e s of c y l i n d r i c a l metal s t r u c t u r e s may prove to be very u s e f u l . The described s t r u c t u r e to be i n v e s t i g a t e d i n t h i s s e c t i o n can be as a c y l i n d r i c a l waveguide loaded p e r i o d i c a l l y with metal d i s c s c o n t a i n i n g c i r c u m f e r e n t i a l s l o t s or as a l i n e a r system of coupled c a v i t i e s . The a n a l y s i s approach i s based upon the coupled c a v i t y viewpoint and i s an extended v e r s i o n of that given by A l l e n and K i n o " ^ ' ^ . This i s not the only cylindrical Two other i n t e r e s t i n g ones- s t r u c t u r e of i n t e r e s t . are the c l o v e r l e a f and the centipede ( m u l t i p l e loop structures"^'"^. coupled) Confluence i n the centipede s t r u c t u r e i s discussed i n Chapter 5. The purpose of t h i s chapter i s to show that a one-mode approximation of the electromagnetic field s t r u c t u r e p r e d i c t s c o n d i t i o n a l confluence s o l u t i o n f o r the and that the i n c l u s i o n of higher modes y i e l d s the same r e s u l t . Also i t i s d e s i r e d to r e l a t e the confluent behaviour of c y l i n d r i c a l s t r u c t u r e s to t h e i r p h y s i c a l c o n s t r u c t i o n . In a d d i t i o n , A l l e n and Kino's theory w i l l be examined i n more d e t a i l than h i t h e r t o presented. 3.2 This examination w i l l be continued A n a l y s i s of the Coupled C a v i t y 3.2.1 The infinite i n Chapter 4. Chain General Theory s t r u c t u r e i s shown i n Figure 19 and c o n s i s t s of an chain of i d e n t i c a l , l o s s l e s s , symmetric c a v i t i e s coupled to one another by long narrow c i r c u m f e r e n t i a l s l o t s . The s l o t s subtend an angle of 2 f and are i n l i n e i-1 Figure i i+1 19, C i r c u m f e r e n t i a l - S l o t - C o u p l e d Structure axially. d<< a C y l i n d r i c a l Slow Wave For the purposes of a n a l y s i s , the c a v i t i e s are numbered; i.e., i - 1 , i , i+lj etc. The w a l l area of c a v i t y i removed to form the s l o t between c a v i t i e s i and i - 1 i s denoted S|« S i m i l a r l y , the w a l l area removed from c a v i t y i to form the s l o t between c a v i t i e s i and i+1 i s denoted SV. The remaining w a l l area i s denoted S^. Before c o n s i d e r i n g the a n a l y s i s of a coupled c a v i t y system, i t i s u s e f u l to examine the procedure f o r determining the non-resonant f i e l d s e x c i t e d i n a s i n g l e c a v i t y by an external source. The f i e l d s can be d e s c r i b e d s p a t i a l l y dependent the p a r t of the resonant mode expressions f o r c l o s e d c a v i t y ( s h o r t - c i r c u i t modes). is e j W i n terms of the The time dependence ^ where (0 i s the frequency of the source. In other words, the f i e l d s are given by e ^ Y ^ e ^n 1" (p) n -i E(p,t) = and n H(p,t) = e J t t t > h H (p) n n n where n denotes one of the resonant modes, f o r example, TMQ^Q. 20 I t has been shown by Teichmann and Wigner , however, that the above s h o r t - c i r c u i t mode expansions are not s u f f i c i e n t to describe the f i e l d s , and that an a d d i t i o n to the magnetic f i e l d expansion, namely the gradient 21 required. Later Kurokawa of a s c a l a r f u n c t i o n , i s showed that the above a d d i t i o n i s not always s u f f i c i e n t and that the c u r l of a v e c t o r i s sometimes r e q u i r e d , the Whichever function of these expansions i s used, s o l u t i o n obtained i s not always v a l i d a t the plane of the coupling holes because these expansions cannot y i e l d a transverse e l e c t r i c the A method f o r determining f i e l d there. expansion c o e f f i c i e n t s w i l l be given belov. In the coupled c a v i t y system, the e l e c t r i c and magnetic f i e l d s of the n'th normal mode i n the i ' t h c a v i t y are denoted E. 1, n and H. r e s p e c t i v e l y and have resonant frequency ft). 1. n 1, n Thus i n the absence of Teichmann and Wigner's g r a d i e n t term or Kurokawa's g r a d i e n t or c u r l term, the f i e l d expressions are E.(p,t) = e ^ " > x ^' ' /_ e . E . (p) j n, 1 n, 1 * ' 1 n (3.1) 1 (p,t) H. = e ^ * ). i hn , i. Hn , i. / (p) * ' n where the boundary c o n d i t i o n s are n. I x E . = 0 n, l on S. + S? + Sl» "n. • H I i . = 0 n, i i I and h\ i s the u n i t normal v e c t o r to the surface d i r e c t e d outward from the c a v i t y . The a c t u a l f i e l d s i n the c a v i t y chain are E.(p,t) = e J w t B.(p) H.(p t) = e J W t H.(p) and f Thus, i n the open r e g i o n E (p) = ^ e ± n i i E n i i (p) n H. (p) =) i N ± w i , n h n,i .H .(p) n,i r / + + S£ 60 Since the normal modes are orthogonal"*", i t i s p o s s i b l e to write e E. . E * dv 1 n ,.i v. l n, I e E. I 2 ,n dv v. I and — _ * u H. . H V . 1 dv n, l I R n.i ix H dv ^ n,i 2 v. I U s u a l l y E^ and IL are known only over the c o u p l i n g and even t h e r e , only approximately. must be converted Appendix I I I . apertures Hence the numerator i n t e g r a l s to surface i n t e g r a l s . This i s done i n The r e s u l t s produced there plus the f a c t t h a t for identical cavities (the Floquet f a c t o r w i l l be i n c l u d e d i n the e's and h's) <0 . = n l n, l + l — etc. = (0 n } E = etc. = E n — etc. = H n, l + l n . = E" n,i H .= n, I n, i+1 us to w r i t e J n 3 e — n,i (» and / 2 -« i — * \ + ) n ' — (\ x Hn ).n.ds ' I (3.2) 2 / V e E" 2 n dv 61 (E h i x H *). n n ± ds (3.3) n,i (<o 2 - <c 2 ) I* H n ' dv 2 n V 3,2.2 Determination of the S l o t The remaining problem Field i s to determine E.. In theory one could express the s l o t f i e l d s i n terms of the set of f u n c t i o n s obtained by s o l v i n g Maxwell's equations f o r the waveguide by the s l o t . By matching the s l o t field formed component to the c a v i t y H0 f i e l d component at the s l o t - c a v i t y i n t e r f a c e s , the amplitude c o e f f i c i e n t s of the s l o t expansion are r e l a t e d to those of the c a v i t y expansion. Combination with the set of equations given by e i t h e r of these r e s u l t s (3.2) or (3»3) will y i e l d a set of l i n e a r homogeneous equations i n e i t h e r the c a v i t y expansion amplitudes or the s l o t expansion amplitudes. The d i s p e r s i o n r e l a t i o n i s obtained by s e t t i n g the determinant of the c o e f f i c i e n t s i n the equations to zero. i s almost e x a c t l y that used i n Chapter Although the above approach long and d i f f i c u l t and Kino who to use. Another This procedure 2. i s v a l i d , i n practice i t i s approach i s that of A l l e n represent the s l o t i n a completely d i f f e r e n t namely, as a f i n i t e way; width p a r a l l e l plane t r a n s m i s s i o n l i n e which i s shorted at 0 = + (|). The e x c i t a t i o n of t h i s l i n e i s by the r a d i a l w a l l c u r r e n t s a s s o c i a t e d with the H0 f i e l d the c a v i t y normal modes. component of These w a l l c u r r e n t s are given by 22 62 J (3.4) = -n x H where h i s as p r e v i o u s l y d e f i n e d . Thus, i n keeping with the above n o t a t i o n , J' . = -n!i x n1 n,I J (3.5) ' v i s the w a l l current i n the i ' t h c a v i t y on the w a l l between the i ' t h and the i - l t h 1 c a v i t y i . The c a v i t i e s due to the n'th normal mode i n t o t a l current i s then (3.6) n To study the t r a n s m i s s i o n d e t a i l , consider Figure the w a l l current r = a-d 20. The l i n e r e p r e s e n t a t i o n i n more e f f e c t of the i component of i s to b u i l d up p o s i t i v e charge on side of the This e f f e c t would be s l o t and the to decrease i t on the r = a s i d e . caused i n a two wire t r a n s m i s s i o n line by a current f l o w i n g from r = a to r = a-d whose magnitude i s given by the c a v i t y w a l l c u r r e n t . s l o t i s due Figure The e x c i t a t i o n of any given to the w a l l currents on both s i d e s . Thus i f i 20, of the S l o t Transmission Line Representation denotes the t r a n s m i s s i o n l i n e e x c i t a t i o n c u r r e n t , the current exciting the i ' t h s l o t , which, by d e f i n i t i o n , cavities i and i + 1 , i s given by ^,1 = ^ i ( ^i+]> + where "u i s the u n i t v e c t o r i n the r r Now i f the Floquet criterion i s between \ direction, i s to be met, the f o l l o w i n g r e l a t i o n s are r e q u i r e d : e . ., = e n,i+l 0 e i n, 1 +1 n,i n, I Thus 3 I^+1 = - n Jl + lx > l_ ,h n,.I,, +1 H n n ? A l J = -n! " xVh Z n -J'^L i n . e - ^ L ii » x n -r, Hence i r f i - <J? * ^ For the purposes of t h i s d << a. II). « R (3.7) study, i t i s assumed t h a t Thus the t r a n s m i s s i o n l i n e equations may be w r i t t e n as and then combined to form \ + It V: = j k Z 6a0 2 1 where k = <n \J LC and Z A 0 r i ' (3.8) X In g e n e r a l , the s o l u t i o n of n (3.8) i s g i v e n by V. = V . ( 0 , h To make equations and n > i ,l//L) (3.9) (3.3) compatible w i t h equations (3.7) (3.4), the f o l l o w i n g changes are r e q u i r e d : (E. x H * ) . n-„ ' I n = -E..(n. x H ) i i n ' — — i J 1 * n,i At SV 6v, E.= u dv r S i m i l a r l y at S| E i u = Thus a$ R .i - i, J"' n , i•*. ur J h . = n,i v - -a(j) V.n ,J i "*..u I J -a^ (« But da0 + / / 2 - <o ) n 2 / n H 2 r dv = e ^ ^ ^ V\ by the F l o q u e t c r i t e r i o n . Hence da0 65 V. j ».*.u d0 + e^ e ^ n,i r 1 -n,1 r L d I v L + / / V. V I j'*..u ''n,! r d0 n.i (tt 2 - tt ) n (i H / n 2 7 2 dv r (3.10) 3.2.3 The Dispersion Relation At t h i s p o i n t , A l l e n and Kino s u b s t i t u t e equation into (3.10) and (3.7). The then s u b s t i t u t e the r e s u l t i n g expression (3.9) into procedure to be followed here i s s l i g h t l y d i f f e r e n t . Equation (3.9) i s s u b s t i t u t e d i n t o equation h . = ) n,i / I ! nm h . m,i (3.10) to produce n = 1,2,3 . ( 3 . 1 l ) m which i s a set of n l i n e a r homogeneous equations i n n unknowns. As p r e v i o u s l y s t a t e d , a n o n - t r i v i a l s o l u t i o n f o r the h's if and only i f det where exists o (I nm - S ) = 0 nnr i s the Kronecker d e l t a . nm dispersion (3.12) This expression i s the relation. 3.2.4 D e r i v a t i o n of a Set of I 's nm The TM, ., and 110* ir set of I nm * s corresponding ° r to the TM-.,,,, TM,.,, , 010 Oil' TM, ,,(TM^ ) normal modes are d e r i v e d . lll 0,r,z' x were chosen because, f o r a /L < ^ l . l j t h e y have the These modes lowest 66 resonant f r e q u e n c i e s of a l l TM mode resonances. A l l e n and Kino confined themselves to a dominant mode expansion (TMQ^Q) and m o d i f i e d 1-^ l a t e r to i n c o r p o r a t e the e f f e c t s of 0 v a r i a t i o n s i n the f i e l d s and of an i r r o t a t i o n a l magnetic f i e l d . The f i e l d expressions f o r the modes above are obtained i n the usual manner and are g i v e n below. Cos 0 was chosen f o r the TM^ and TM-j^^ mode expressions because i t was f e l t that the E^ field component would be e x c i t e d w i t h a maximum a t 0 = 0* a) ™ Q 1 0 ~ normal mode one (n = l ) E = J z (K r) Q x J ^ r ) H 0 = ~ 3 where (3.13) K l E b) 2.4048 a = ^1 = =E =E r r d Y> c =0 TMQ-^ - normal mode two (n = 2) E = J (K 0 z r E H i r )sin = " ? J L l J 0 = J ( K l > i<V? where K, = l a 2.4048 a 2 tt E 0 5 H r 3 0 r c o s i n 2 s T ( 3 ' 1 4 > 67 c) ^110 ~* E H z 0 = n o r J l a m ° d e three (n = 3) l( 2 ) K r c j« 3 K„ ^— r o ^ s J\(K r) e 9 ~xr = H m = cos 0 J (K r ) s i n 0 (3.15) where 3.832 a 2 E d) r = E0 s *3 0 ™ ; Q I "" normal mode four (n = 4) E = J ^ ( K r ) cos 0 s i n — j - E_ = 2 J (K.r) 7 Tt K L 2 E 0 = ftK -Lr 2 J l ( K 2 r ) s i n 0 c o s 0 cos cos w TtZ (3.16) 68 H H =- 30 4 K 2 e (K^r) s i n 0 s i n r J (K r) 3»4 e 0 ~ " ^ x K, J 0 2 > ( K ~ r 2 0 cos W s i n K„r 3tz where '40- (0 3 1 2 I . c 71 + L The s t o r e d energy expressions f o r the i n d i v i d u a l normal modes are a) TM 0 1 0 : u ^ b) TM 0 1 1 2 dv = TM 1 1 0 E i n \m a > 2 2 e ~ 2 La J 2 2K^ 2 1 ( K a) 1 P> 2 : H d) V : u HL dv c) J^C^a) dv ueLa J^ (K a) 2 2 2 : (i H A dv = |ATt w 2 A e La 2 4K, 2 jQ (K a) 2 2 J (3.17) 69 Equations (3*6) and (3.7) permit the f o l l o w i n g truncated r a d i a l w a l l current expression to be w r i t t e n : i f . ( r = a) = 3 h l f © + 3h + 3h . i e J^^a) _ 2 2,i 1 -a> e J ( K a ) ' 3 Q 2 3,i -y + j h 4,i A h l , i + B J h o ( K 2,i + 2 - ^ e ^ + i L L - cos 0 a ) ( C 1 h 3,i e " ^ + + D h cos 0 L 4,i^ c o s ^ Thus from equation (3o8), 6a0 x 2 - k Z A h, . + B h_ . + (C h- .+ D h. .)cos 0 0 JLyX & y1 _? y 1 4 j l cos 0 = A' + C w i t h boundary c o n d i t i o n V\ ((|J) = V\ (-(])) = 0. The s o l u t i o n of t h i s equation i s V. = A" cos ka 0 + ±1 + '*V° k^ k Lk C where A' A": 2 C a cos (j) ,22 , k a -1-1 0S (3.18) 2 cos ka(i} a - 1 (3.19) 70 S u b s t i t u t i o n of equations (3.17), (3.18) and (3.19) i n t o equation (3.10) and determination of j " . and j ' . II f 1 11)1. by use of expressions (3.5), (3.13), with subsequent s u b s t i t u t i o n i n t o homogeneous equations (3.1l). (3.14), (3.15) and (3.16) (3.10) leads to the set of The s u b s c r i p t i may now be dropped because r e l a t i v e p o s i t i o n i n f o r m a t i o n i s no longer required. The expressions f o r the I -2 B (tan - QAn/2 11 £l(D. fiAn/2) 12 H(H J (K a) 0 2 Bp ^(cos l ( K l a = 2 1 '22 2 2 (tan DAn/2 J (K a) 23 2 - J (K x $)sin l//L/2 2 - D ( ^ - 1) 2 - pfisin (p) sin\jjL 2 i a ) - l)(Sl - 2 2 1) 'sin\/;L u^r) -DK%/2) , f t ( f t 0 -p£lsin <|j tan flAn/2 (p Cl _ i^/^ma 2j sin l//L (p n -2,j B (tan £2 Arc/2 -QAn/2) -4B sin XJJh/2 ) V j found to be e (|) t a n QAn/2 2 (cos J r - i) 2 J^K^a) = -o a [tan£2ATi/2 - QAn/2 j B 14 s - 1) 2 13 ' n m cos ^L/2 2 , 2 -( t t ^ V ) 2 ) f ) sinl^L -( « p,Qsin cos $ tajiQA'rc./2. ( p 2 a 2 71 J ( K a ) • Q p ( c o s $ tanaATt/2 - p,Qsin $ ) ( V S ) cos li;L/'2 0 I = 2 4 4B 2 2 ( ^ ) ( ^ ) ( p <d J ^ a ) hi = 4B 2 2s, *3< p J ^ a T n I 3 2 ) - D(a* a 2 -n m 2 $ ian.OATi/2 - p H s i n (^)Bp n(cos -( « 2 T 3 sinv/;L ) = -2j ( p 2 ^ - 1)(,Q 2 (^/tt,) ) - 2 2 2 p £1 cos $ tan £2 Art/2 - <J> ( 0 , Q 2 L - 2 B P 33 3J n2 n ,3n2 J ^ 2 ,( ^V) - ^ ( P .- ( p n H 2 2 4j k J ^ a T 2 - 1) + 1 ) sin n 2 Bp n(^-)(cos 2 tanfiAn/2 2 ( ^ ) ( .2/ p 41 ! 2pHcos $ 34 L - 1) 2 2 ^(^a) 2 - (p n + i) sin $ cos (j). s i n ^ L / 2 p 2 n 2- D ( a -( 2 ( a B p - 2 ( p i a 2 H (cos ^,tannAjl/2 - p£>sin $) s i n l/;L/2 2 2 J (K n 2 2 cos (fl. s i n \JJL - - u^)*) $ ta QATx/2 - p , Q s i n 2 s±n\jjL n> C^){p a 2 2 - 1) (rf- (<* /*l' ' 2 4 W 72 tt*, 6) o ( ^ ) ( ^ ) p Q ( c o s f tan QATt/2 - p f t s i n f ) cosfyL/2 2 J ^ a f j I 4 2 = 8B l J (K a) 0 2 l > to ) 2 p n c o s $ t a n £lAn/2 2 - 2jB p f o ( £ ) 2 •43 -<P ( P ^ 2 - (p Q, 2 2 - 1) + 1 ) s i n $ c o s (j) 2 £)< p n - D (n 2 2 2 2 pHcos^ 2 ( p n .2 - 4B p "44 - 3 ^ - 2 2 a 2 4 / « i ) 2 ) s i n (J> c o s <I), c o s ^ L / 2 (VV^- ft>/» where (3.20) 1 A o» V" 1 S = %</>^/2p^ = f i r s t ai s l o t resonant $ = 0.65319A p = 2.4048 Z a Q B = P 7?L 3 J ^ ^ a ) = 0.5191 J ( K a ) = - 0.4028 0 2 tt, jl + 1.7066 ( f ) 2 tt. 1.5934 tt tt n > 1) 2 - 1 ) ^ (Of- - ( « t a n £lA%/2 s - ( p a + l) 2 ( P 2 sin a\2 1.5934 ,1 + 0.67212 (jr) frequency The I's above correspond to a s t r u c t u r e i n which the s l o t s are a x i a l l y i n l i n e . However, i t i s i n t e r e s t i n g to compare d i s p e r s i o n curves f o r t h i s s t r u c t u r e with those f o r a s t r u c t u r e i n which the s l o t s are r o t a t e d 180° d i s c , as shown i n Figure 21• S t r i c t l y , t h i s structure i s i-1 Figure 21. i C y l i n d r i c a l Structure p e r i o d i c over length 2L, but we the p e r i o d i c l e n g t h provided we properties. structure a) from d i s c to i+1 with Rotated S l o t s can analyze i t assuming L to be enforce c e r t a i n f i e l d symmetry These f i e l d symmetries are a consequence of symmetry and In the are the : rotated slot structure E ( r , 0 . z + L) = E ( r , 0 - n, n whereas i n the non-rotated s l o t n Z) structure E ( r y 0 , z + L) = E~ (r,0, z) n n Similarly ± Further d i s c u s s i o n of the f i e l d symmetries r e s u l t i n g from s t r u c t u r e symmetry w i l l be presented i n Chapter 6. H ( r , 0 z + L) = H (r,0 - Tt, z) ? n b) n The s l o t V s are now r e l a t e d as V (0) i+1 = e ~ ^ V.(0-TX) L As a r e s u l t of these changes, C i n equation 1 a> e J (K ,a) 3,i K 3 k Z0 Q 2 + , .« e -j\//L J (K a) 4 p (3.18) becomes _ 2 ^ + h, since ( r ' 0 ) = J ' n»i - » ~ ( r 0 % ) and h _ = e - ^ n,i+l L h i Also the f o l l o w i n g h o l d s : -Tt+$ * — — j'n. ' 1. » E1 ds = — a -Tt-$ S! I -Tt+$ = - a / e il//L V (0+n) i (0). u -TTU = - a . ^ L J v i ( 0 ) (0—re), u, -0 The f i n a l r e s u l t of these t r a n s f o r m a t i o n s i s that the new I denoted I , are nm' 1 75 1 ll - I " 11 1 A I' - I 12 " 12 x —.1 I 13 ~ s m ~, 2,|, 2 sin l|;L/2 2 13 T / o -2 sin l//L/2 j 2 I' 1 4 •=J ~inl//L s i n = I' 21 1 4 - I ~ 21 x I' 22 =1 22 t _ 2 cos lL>L/2 j 23 , '23 J sm 2 T = l/lT j sinlj 24 X I 24 2 cos V JI./2 2 isin " 2 sin l/,L/2 3 1 2 TT 3 3 1 _ -2 c o s ^ L / 2 2 2 T -0 sinl/,L = 3 2 _ cossl/^L/2 j 2 r = 33 ~ _j_2,/ T t X - 34 ir, "33 s i n 1//L/2 _ I T J T w. x 34 2 sin l/;L/2 . . J„ 41 2 X 41 1 3 -j sin II 4 T t x s i n 2 cos l/;L/2 2 43 2 4 2 T " 43 s i n \bh/2 2Z/ oi ) 2 T 1 44= 9 COS \jJh/2 T J 44 { 3 m 2 1 ) 76 3.3 One-Mode P r e d i c t i o n of Confluence The work of A l l e n and'Kino i n d i c a t e s that the f i r s t c a v i t y passband ( e x c i t e d c a v i t i e s coupled together by s l o t s ) due to the TMQ-^Q mode can o f t e n be d e s c r i b e d reasonably w e l l by the dominant mode d i s p e r s i o n r e l a t i o n ( I - Q - 1 = 0 ) a/L i s not too l a r g e * provided I t a l s o i n d i c a t e s that i f 1 ^ i s modified to include the e f f e c t s of an i r r o t a t i o n a l magnetic field, t h e o r e t i c a l and experimental d i s p e r s i o n curves correspond much b e t t e r f o r both the f i r s t c a v i t y and the f i r s t ( e x c i t e d s l o t s coupled together by c a v i t i e s ) . s l o t passbands The m o d i f i c a t i o n c o n s i s t s of a change i n the s l o t resonant frequency, change i n the c o u p l i n g j and a c o e f f i c i e n t , B. In view of these r e s u l t s , i t becomes meaningful to see i f the dominant mode d i s p e r s i o n r e l a t i o n w i l l p r e d i c t Since the c o u p l i n g confluence. i n the s t r u c t u r e being discussed i s p r i m a r i l y magnetic, the f i r s t two passbands have the f o l l o w i n g form of d i s p e r s i o n curve: Figure 22. A T y p i c a l D i s p e r s i o n Curve f o r a C i r c u m f e r e n t i a l - S l o t Coupled C y l i n d r i c a l Structure 77 The dominant mode d i s p e r s i o n r e l a t i o n i s 2B s i n 2 4>L/2 = ^ ^ tan £lh%/2 In order f o r (3.22) ^ATC/2 to be z e r o , e i t h e r £1 must equal one >QAn/2 = ( 2 n - l ) u / 2 ; i . e . , <o = <o , <o S-L , etc* or Thus a p o s s i b l e S c o n d i t i o n f o r confluence i n t h i s case i s fi>, = A) , which i s 1 si satisfied i f $ = n/2p . Confluence at the zero—mode w i l l l^L = 0 i f , under the c o n d i t i o n above, the group v e l o c i t y at is exist non—zero* Before f u r t h e r c o n s i d e r i n g zero-mode confluence, i t i s of i n t e r e s t to e s t a b l i s h t h a t the one-mode approximation does p r e d i c t confluence between the c a v i t y passband and the second passband at the u'-*mbde i f ] < * 2 6, < s > If H > 1 , then -fiAit/2 > TJ- . Thus ( f i A n / 2 ) of F i g u r e 23o Region passband and Region passband» 2 (tan i7Ail/2 - J7Ait/2) > 1 and A > 1, must be in: e i t h e r r e g i o n I or I I I (© > %/2) corresponds to the II (0 >3%/2) corresponds to the second But t h i s i s impossible since as © © < 3it/2» tan^2Ait/2 — + TlA%/2) cannot oo cavity *- 3n/2 but In other words, con- passbands i s impossible i f < fl) s 2» c o n s i d e r the p o s s i b i l i t y of zero—mode confluence. c o n d i t i o n to be a p p l i e d i s a>-^ = <i>^ o r , i n other words, A = 1. slot which means t h a t (tan^2Alt/2 - remain l e s s than z e r o . f l u e n c e of these two The Also f o r n For it-mode confluence to occur, r e g i o n I I I must disappear. Now passband H ( l - £l ) < 0 and hence, i n order t h a t a r e a l value of \ph e x i s t , must a l s o be l e s s than z e r o . s Now slot (which can only occur i f because the h i g h e s t frequency of the f i r s t o c c u r s : at \jj\h = 0 ) . not Pigure 23• 2B s i n 2 \pL/2 = Pass and Stopband Regions tan "^ 2 ) (3.23) Hence s i n XPL/2 = ± " 2B(tan nArt/2 - QAn/2) and cosl//L/2=' /2B(tannATt/2 - D,An/2) 2B (tan ,Q A n / 2 - - H ( l -,Q ) flAn/2) 2 79 The p o s i t i v e square root i s taken f o r cos ^ L / 2 and both p l u s and minus signs are r e t a i n e d i n the s i n \ph/2 f u n c t i o n we are i n t e r e s t e d i n the behaviour of near because \jj = 0. I f e q u a t i o n (3«2«3) i s d i f f e r e n t i a t e d with r e s p e c t to \p and, i n the r e s u l t i n g expression, s i n \jjL/2 and cos \Jjh/2 are r e p l a c e d by the f u n c t i o n s on the previous page, the f o l l o w i n g results: [A(H)+B(fi)] j£Jj = ± h ^ c ( H ) [ D ( H ) - C(^)] where A AO BtQ) = i 3a = i - = 2 -u/2)Q ( I-n )a 2 cos fln/2 •c(4D = na - cos nu/2) (sinflit/2.- 2 ((In/2) cos,Qix/2) -a ) 2 - £i%/2) D(ft) = 2B The p o i n t of i n t e r e s t i s ,0, = 1 where A(l) = - 2 B(l) = ? C(l) = 0 D ( l ) = oo The f o l l o w i n g l i m i t s are determined through use of l ' H o p i t a l ' s rule: lim DCft) C(Q) a — 1 l i m B(Q) = - 2 a — 1 = S| 80 Thus • or (3.24) The dominant mode approximation d e f i n i t e l y i n d i c a t e s that the c i r c u m f e r e n t i a l - s l o t - c o u p l e d c a v i t y system can he.made c o n d i t i o n a l l y confluent at the zero-mode by making the f i r s t s l o t resonance occur at to = confluent . The group v e l o c i t y at the zero-mode i s double signed because the d i s p e r s i o n curves cross there as shown below. in — ^ L i — r —TC Figure 24. Zero-Mode Confluence I t has been shown thatft), = 0), i s , according sl 1 ' to the one-mode a p p r o x i m a t i o n , . s u f f i c i e n t to produce confluence. It w i l l now be shown that i t i s a l s o a necessary c o n d i t i o n . I f 81 the product s i n lph/2 cos \ph/2 i s r e t a i n e d when equation i s d i f f e r e n t i a t e d , a s l i g h t l y d i f f e r e n t expression (3.23) f o r d,Q/d^; r e s u l t s ; namely, F(ft, A ) + dQ ^ffj G(H,A) = 2 BL 0 m . \JJL l k sin ^ cos where (sin 0-0 cos 0) and 9 = £lA%/2 Since t h i s argument concerns confluence between the f i r s t passband and the f i r s t ( A < 3). c a v i t y passband, tt^ must be l e s s than But f o r A / 1, the f u n c t i o n e i t h e r tt = tt^ or tt -^. g A = 1 i s (F + G) zero. ment can be w r i t t e n : predicts G Thus (F.+ G) i s not zero must be zero. c a v i t y passband and the f i r s t w s i * e f f e c t of the i r r o t a t i o n a l magnetic f i e l d above i s not immediately c l e a r . Allen"^ - L n ( ji c a ^ e s state- A l l e n and Kino's one-mode d i s p e r s i o n r e l a t i o n confluence of the f i r s t The Only when Consequently the f o l l o w i n g general s l o t passband at ipL = 0 i f and -only i ftt^= the tt 2 When \JJL = 0, the r i g h t hand side of the equation above i s zero. for slot on the r e s u l t As stated e a r l i e r , the work of that i f i t i s taken i n t o account, a change i n s l o t resonant frequency and i n the c o u p l i n g f a c t o r , B, r e s u l t s . This change i s \Jj independent i n the r o t a t e d s l o t s t r u c t u r e but not i n the non-rotated case. In A l l e n ' s approximate a n a l y s i s , the dominant mode d i s p e r s i o n r e l a t i o n i s f o r m a l l y the same as except (3.22) 82 B — B yjji - C cos where i n the r o t a t e d s l o t case C — 0 . l/^L = B» I t i s shown i n Appendix IV that i f the f i r s t mode d i s p e r s i o n r e l a t i o n i s a l t e r e d i n the above manner, the group v e l o c i t y a t the c o n f l u e n t zero-mode p r e d i c t e d by i t i s where A = 1 + (a term due to the i r r o t a t i o n a l f i e l d ) , must be a p o s i t i v e non-zero (A-C) number otherwise the e f f e c t i v e c o u p l i n g c o e f f i c i e n t , B ' , and the e f f e c t i v e s l o t resonant frequency are no longer meaningful. 3.4 Multi—Mode Dispersion Relations F o r t r a n programs f o r computing the d i s p e r s i o n curves pro- duced by the 3x3 matrices r e s u l t i n g from the f i r s t three normal modes d e s c r i b i n g the r o t a t e d and non-rotated s l o t s t r u c t u r e s were written. U n f o r t u n a t e l y they e i t h e r d i d not p r e d i c t continuous curves or e l s e p r e d i c t e d curves which d i f f e r e d enormously those g i v e n by the dominant mode d i s p e r s i o n r e l a t i o n . from While attempting to f i n d the reason f o r these r e s u l t s , i t was found that the 2x2 matrices r e s u l t i n g from the equations l i s t e d below produced a) continuous d i s p e r s i o n curves: TM (I i b) and T M Q 1 0 - l)h u 3 \ 1 (i I TM N L 2 + ( I 3 h 3 =• 0 - l)h = 0 ^ - s l o t s not r o t a t e d 2 h l 3 3 3 - i)h + i 2 2 4 2 + I n + (i TMQ.^ and - s l o t s not r o t a t e d 1 1 0 44 " 2 l ) h 4 h 4 4 = = 0 0 (3.26) 83 TM and T M 0 1 1 (l 2 2 l^ 1) h r 2 h 1 1 - s l o t s r o t a t e d 180 + I- h 2 3 + (l^ 2 0 = 0 3 - 1) h 3 = 0 3 I t was a l s o found that the f o l l o w i n g combinations gave anomalous results: a) TMQ^Q and TMQ-Q - s l o t s both r o t a t e d and not r o t a t e d <I -1) n 21 b) TM (I Q 1 0 11 h l + h ( I and ~ l h 31 } 1 22 " TM x 1 l h I + + 1 I 1 } h 2 h„ = 0 „ 2 2 - ° - s l o t s r o t a t e d 180 0 13 + (i^ 1 = h 3 (3.27) 0 - 1) h 3 C 3 = 0 Some numerical r e s u l t s produced by these equations are shown i n F i g u r e s 25 to 28. F i g u r e s 25 to 27 r e s u l t e d from The (3.26). s o l i d curves are from A l l e n and Kino's dominant mode a p p r o x i - mation and the dashed curves are from the two-<mode approximation. I n c l u d i n g the TM^ Q mode i n the theory has q u i t e a d r a s t i c effect N on the upper ( c a v i t y ) passband. These r e s u l t s do not i n c l u d e the e f f e c t of the i r r o t a t i o n a l magnetic fields, According to A l l e n and Kino *s procedure, i t s i n c l u s i o n i n the multi—mode a n a l y s i s leads to the f o l l o w i n g e x p r e s s i o n s : "11 1 3 = 2 ft(ft J (K a) 0 I _ -2B' (tan 0' 2 B« pm 2 Oil 2 3 1 =4 2 2 p* Q s i n 2 f t a n ©' - -(<*J« ) )(p n l - i)(p n B'p n(cos (H t a n ©' 2 2 I - 1) cos (n ^L/2 sin 2 2 2 - D) p*Si s i n $) s i n l ^ L / 2 2 - D) 84 > o Ol O * 00 • \D 1.4" 86 S l o t s Rotated 88 2 p* 57 c o s - $ (p' H 2 -2B' p f i 3 i/7 [- ( p ' , Q 2 2 '33 ft - (jr ) 2 1 (P a 2 2 2 2 2 $ tan ©' - 1) +1) sin $ - D) cos si n \pL/2 2 : 1 where B P' = p ©' = D The per /A / V/A ^7A' = A - C cos I p L ' - C cos l / / L n/2 = C Q A T C / 2 ) / Jk - C cos \//L - C cos r e s u l t s above i n d i c a t e that e i t h e r more than one s e c t i o n or r o t a t e d s l o t s should be used i f the slot s t r u c t u r e i s to be employed i n a t r a v e l l i n g wave tube f o r which a large bandwidth is desired. However, the presence of the TM-^Q mode does not a f f e c t the shape of the d i s p e r s i o n curve i n the neighbourhood of a confluence p o i n t . This i s shown i n Figures 26 and 27 where the two- mode d i s p e r s i o n curves c o l l a p s e to the one-mode curves as \pL *- 0« At the confluence p o i n t h^ i s zero; i . e . , the TMQ^Q mode dominates. Figure and was 28 i s a d i s p e r s i o n curve f o r a r o t a t e d s l o t computed using (3.27). structure Note that the two-mode c a v i t y d i s p e r s i o n curve i s almost the same as that given by the dominant mode d i s p e r s i o n r e l a t i o n . However, the s l o t passband r e s u l t s are completely anomalous i n t h a t they do not p r e d i c t a s i n g l e valued curve i n the range 0 ^ \ ^ L ^ TC. 89 The i n c l u s i o n of the TM^ ^ mode i n the theory f o r non- n r o t a t e d s l o t s t r u c t u r e s y i e l d s r e s u l t s which d i f f e r g r e a t l y those given by the dominant mode expansion. i n v e s t i g a t i o n was An experimental c a r r i e d out to t e s t the expanded theory the r e s u l t s are presented i n the next from and chapter, 3. 5 The E f f e c t of an A d d i t i o n a l N o n - C r i t i c a l l y - R e s o n a n t Coupling Element on Confluence In any s t r u c t u r e to be used i n an a c c e l e r a t o r there must be a hole through the center of the l o a d i n g d i s c to permit sage of the e l e c t r o n beam. minimized pas- Coupling through t h i s hole can be by use of a d r i f t tube. However, i t i s i n t e r e s t i n g to consider the e f f e c t of an a d d i t i o n a l non-critically-resonant coupling element on the confluence c r i t e r i o n . To t h i s end, i t i s perhaps more appropriate to consider a s t r u c t u r e with an a d d i t i o n a l long c i r c u m f e r e n t i a l s l o t than one w i t h a c e n t r a l hole. Not only i s the m u l t i p l e - s l o t case more amenable to analy- s i s , but s l o t s are much stronger c o u p l i n g elements. The l o a d i n g d i s c of the s t r u c t u r e to be analyzed i s shown i n Pigure 29. In a one-mode a n a l y s i s where the i r r o t a t i o n a l magnetic mode i s ignored, the e f f e c t of c o u p l i n g between the two i n one l o a d i n g d i s c i s not represented. of the behaviour of one slots However, p e r t u r b a t i o n s l o t by the other should be minimal i f the d i f f e r e n c e between the s l o t resonant f r e q u e n c i e s 90 (a> sl Figure 29. and « ) i s large. g l Loading Disc f o r a M u l t i p l e - S l o t I f f o r both a> the theory p r e d i c t s confluence, intermediate - ©' gl g l Structure l a r g e and 6 s s i = W then the s l o t - s l o t coupling at p o i n t s can be expected to produce a c o n t i n u i t y of the group v e l o c i t y a t the confluence p o i n t as a f u n c t i o n of A one-mode a n a l y s i s of a s t r u c t u r e with an a d d i t i o n a l s l o t y i e l d s the f o l l o w i n g d i s p e r s i a n r e l a t i o n : 2B s i n -SI ) 2 = (tan If sl - + (tai±nA it/2 2 - ftA */2) 2 A-^ i s set equal t o u n i t y and the d i s p e r s i o n r e l a t i o n d i f f e r e n t i a t e d w i t h respect to \jj , the r e s u l t i s which i s the same as given by the one-mode approximation f o r The confluence c r i t e r i o n , A ^ = 1, the o n e - s l o t s t r u c t u r e * i s unchanged. If = <©*^ = »^, g - 1 then ll it I t i s to be expected t h a t the e f f e c t of the i r r o t a t i o n a l mode on the group v e l o c i t y a t the c o n f l u e n t zero-mode i s to make i t a continuous f u n c t i o n of A where the f u n c t i o n f ( A ) 1 2 S 2 * I n other words s i m i l a r to that g i v e n by A l l e n and Kino f o r the i r r o t a t i o n a l mode. F i g u r e 30. presents a g r a p h i c a l i l l u s t r a t i o n of some of these r e s u l t s . are used i n a c o n f l u e n t resonant If n identical slots s t r u c t u r e , the frequency s e p a r a t i o n between the 1J4 = 0 res onance and i t s nearest neighbour i s i n c r e a s e d by approximately single-slot case. Figure 30. i l l u s t r a t e s t h i s f a c t f o r the two- slot case. 3.6 Concluding Remarks Since from t h a t of the mode separation*' i s the important 1 p r o p e r t y of c o n f l u e n t s t r u c t u r e s$. i t i s very d e s i r a b l e to know how dependent i t i s on whether or not the confluence met. An i n d i c a t i o n criterion i s of t h i s dependence can be obtained by p l o t t i n g £1 f o r \|/L/u .= 0»05 (i«e., a 2 0 - s e c t i o n c a v i t y ) as a f u n c t i o n of A i n the case of dominant mode One such p l o t i s given i n F i g u r e 31,« This term i s d e f i n e d i n Chapter 1 excitation* This curve i s f o r 93 1.02 1.04 1.08 1,06 si Figure 31. Mode Separation Cavity as a Function a s t r u c t u r e which i s almost confluent shows that mode s e p a r a t i o n of A f o r a 20-Section a t the zero-mode and does not suddenly change as the confluence c r i t e r i o n i s s a t i s f i e d . The same behaviour has been 4 observed t h e o r e t i c a l l y i n a d i e l e c t r i c loaded s t r u c t u r e terms of the d i s p e r s i o n curves, the f u n c t i o n \& n/a\p 2 • In 2 1 at IJOL = 0 becomes l a r g e r as the confluence c o n d i t i o n i s approached. Thus i n p r a c t i c e i t i s not necessary to have an e x a c t l y s t r u c t u r e to achieve good mode separation; an almost confluent confluent s t r u c t u r e w i l l do. The theory presented i n t h i s chapiter has been f o r l o s s - less structures. One might wonder i f the presence of l o s s i n a s t r u c t u r e w i l l a l t e r the confluence p r o p e r t i e s found f o r l o s s less structures. To consider t h i s problem two analyses were performed assuming l o s s to be present i n the s t r u c t u r e . They are presented i n Appendix V. In the f i r s t transmission line a n a l y s i s i t was assumed that the s l o t contains series loss. Equations (V*-l) and (V-=«2) then d e s c r i b e the d i s p e r s i v e c h a r a c t e r i s t i c s of the s t r u c - ture. 6>^L/r = 1000, these equations p r e d i c t that I f one l e t s the mode s e p a r a t i o n of both a 10 and a 20—section confluent c a v i t y i s decreased by l e s s than lfo by the i n t r o d u c t i o n of l o s s . However, the group v e l o c i t y at the c o n f l u e n t (see F i g u r e 32(c)). In order to introduce the zero-mode i s zero l o s s i n t o the c a v i t y as w e l l as s l o t j an a n a l y s i s of a lumped e q u i v a l e n t c i r c u i t f o r the c y l i n d r i c a l slot-coupled s t r u c t u r e was performed. ( V - l ) and (V-3) d e s c r i b e the d i s p e r s i v e c h a r a c t e r i s t i c s i n t h i s case. I f Q,, the Q of the c a v i t y , and Q Equations the Q of the s l o t , are oo and 1000 r e s p e c t i v e l y , the d i s p e r s i o n r e l a t i o n p r e d i c t s almost e x a c t l y the same r e s u l t s as p r e d i c t e d by (V^-l) and (V-2) with n^/r = 1000. curves i n F i g u r e If Q ± = 5000, Q g l 32(b) and (c) r e s u l t . = 1000 and A = 1, the The extreme upper and lower s t r a i g h t l i n e s are the d i s p e r s i o n curves p r e d i c t e d when Q n **1 = Q , = oo . *sl The others are f o r Q, = 5000 and Q , = 1000. 1 sl These v a l u e s were chosen t h i s low to account f o r most p r a c t i c a l cases. I t can be seen t h a t the d i s p e r s i o n curve f o r the l o s s y s t r u c t u r e p a r a l l e l s that f o r the l o s s l e s s s t r u c t u r e except near i f l = 1.0. As fl 1.0, the d i s p e r s i o n curve f o r the l o s s y s t r u c - ture approaches ^pL := 0 l i n e a r l y but w i t h l e s s slope r other d i s p e r s i o n curve. Again there the mode s e p a r a t i o n of c o n f l u e n t than the i s no s i g n i f i c a n t ^ c h a n g e i n 10 and 20—section cavities. 95 1.0005 a 0.9995 1.00005 0.0001 0.0001 Figure 32. D i s p e r s i o n Curves f o r a Lossy C y l i n d r i c a l S l o t Coupled S t r u c t u r e If - Q i = s 1000, the group v e l o c i t y at the confluent zero—mode i s the same as f o r the l o s s l e s s s t r u c t u r e . attenuation i s present (a; ^ 0) as one would expect. However, Thus the models used f o r these analyses i n d i c a t e t h a t the e f f e c t of l o s s on the d i s p e r s i o n p r o p e r t i e s of a confluent cylindrical slot- coupled s t r u c t u r e i s dependent upon the r e l a t i v e values of the l o s s i n the c a v i t y and the l o s s i n the coupling element. In p a r t i c u l a r , i f one wishes to o b t a i n a l a r g e group v e l o c i t y at the confluence p o i n t , the a n a l y s i s above i n d i c a t e s that the l o s s e s o f the c a v i t y and s l o t should be about Unfortunately, equal. the zero-mode confluence d i s c u s s e d i s not u s e f u l i n an a c c e l e r a t i n g s t r u c t u r e * of the: zero-mode confluent above The \p= 0 harmonic s t r u c t u r e has i n f i n i t e phase v e l o c i t y 96 and the other space h a r m o n i c s , w h i c h do have f i n i t e velocity, have small amplitudes. structure a t \jj = 0 i s o n l y s l i g h t l y T M Q ^ waveguide mode* In fact, phase t h e wave i n t h e different from t h e c u t - o f f For instance, the r a t i o o f the amplitude o f t h e \fjh = 2TC h a r m o n i c t o t h a t o f t h e =0 harmonic i s g i v e n b y (see A p p e n d i x V l ) fl a _ sin % £ /I 0 1 + £ TC where ^ i s t h e t h i c k n e s s o f t h e c o u p l i n g w a l l s . £ — O j a l^ 0 * a one a > o r / Q a n v a r a ^- l goes t o z e r o 0 values o u s not been found obtained f o r a l l n ^ 0. i s small. the structure treated i nt h i s t o have a u s e f u l c o n f l u e n c e , i nthis A table of o f ^ / L i s g i v e n i n A p p e n d i x vit where c a n see t h a t t h e r a t i o Although into s In fact, i f study provide one w i t h t h e phenomenon o f c o n f l u e n c e . seen t h a t t h e c l o v e r l e a f chapter has the results considerable insight F o r i n s t a n c e , i t c a n be and t h e i n t e r l a c e d o r c e n t i p e d e 18 structures are good/prospects B o t h have r e s o n a n t indirectly, frequency ability c o u p l i n g elements which, d i r e c t l y or i n t r o d u c e a TC phase s h i f t . I f the resonant o f the c o u p l i n g elements o f these made e q u a l should f o r use i n resonant a c c e l e r a t o r s . t o t h a t o f t h e d o m i n a n t c a v i t y mode, -jc-mode c o n f l u e n c e occur. The c e n t i p e d e t o support interesting two s t r u c t u r e s i s s t r u c t u r e , which has t h e proven 23 high f i e l d s , i s p e r h a p s t h e more o f t h e two s i n c e i t h a s a x i a l d i s c u s s e d f u r t h e r i n C h a p t e r 5» symmetry. I t w i l l be 4. AN INVESTIGATION OF THE ACCURACY OF THE TVO-MODE ( T M TM 1 1 0 0 1 0 - ) DISPERSION RELATION In view of the l a r g e d i f f e r e n c e i n t h e o r e t i c a l l y p r e d i c t e d d i s p e r s i o n curves f o r the s t r u c t u r e with s l o t s i n l i n e and the s t r u c t u r e w i t h s l o t s r o t a t e d , an experimental investi- g a t i o n o f the accuracy of the p r e d i c t i o n s was d e s i r a b l e . Accord- i n g l y a f o u r - s e c t i o n c a v i t y with the f o l l o w i n g dimensions was built: a = 0.532 i n . = 1.35 cm i L = b+t = 0.293 + 0.016 = 0,309 in. = 0.785 cm d = 0.082 i n . = 0.202 cm (j) = 56.2° (one s l o t / d i s c ) where t i s the thickness of the d i s c s . The t o l e r a n c e s where - 0,001 i n , . The c o n s t r u c t i o n d e t a i l s are shown i n Figure 33. The coupling w a l l s were d e s i r e d to be as t h i n as p o s s i b l e y e t still t h i c k enough to maintain t h e i r shape. 0.016 i n . t h i c k . They were made Four s e c t i o n s were b u i l t because i t was f e l t that 5 d i s p e r s i o n p o i n t s were adequate to determine how the t h e o r e t i c a l d i s p e r s i o n curves correspond to the experimentally determined ones. The c a v i t y was assembled w a l l and i n s e r t i n each of the center r i a t e end p i e c e . by p l a c i n g a coupling s e c t i o n s and the approp- The s e c t i o n s were then stacked one on top of the other and placed between the end p l a t e s which were then b o l t e d together. Input and output coupling to the c a v i t y was through small probes, one i n each end w a l l . The value of $ above corresponds to **^/ ^ a s = A =1*5 which was chosen t h i s l a r g e to ensure that fi>,/fl> > 1.0, The ° 1 s i l e n g t h of each s e c t i o n , L, was chosen t o be small enough to give , 1 98 F i g u r e 33, C o n s t r u c t i o n Drawing of the C y l i n d r i c a l a reasonably Cavity l a r g e value to the coupling c o e f f i c i e n t , B, a l s o l a r g e enough to keep the f a c t o r C i n expression but (3.25) rela™ 17 t i v e l y small (see A l l e n and Kino )» The reason f o r t h i s last c r i t e r i o n f o l l o w s from the f a c t t h a t , according to theory, C gives r i s e to an expansion of the c a v i t y passband whereas the ^110 m o ^ e < l o e s "the o p p o s i t e . m e n t i t i s necessary For the purposes of t h i s e x p e r i - that the e f f e c t of the TM-^Q mode dominate. Assuming the t h i c k n e s s of the d i s c s to be zero, ZQ/T^ can be c a l c u l a t e d by u s i n g the i n v e r s e cosine conformal t r a n s 24 formation to f i n d the d i s t r i b u t e d capacitance of a t r a n s m i s s i o n 99 l i n e c o n s i s t i n g of a small gap between a h o r i z o n t a l plane and a p e r p e n d i c u l a r s e m i - i n f i n i t e plane* infinite The capacitance i s r e l a t e d to ZQ as z o _ IT yc ftc c 1 1 1. ~ cc 16 IT This i s the procedure used by A l l e n and Kino * . F o r the s t r u c t u r e d e s c r i b e d above, we get Z 0 Z jj- = 0.302 and B = 0 a 4 ^ = 0.05 The c a v i t y was designed to permit any d e s i r e d angular o r i e n t a t i o n of the s l o t s . The hole ,(1/16" d i a . ) p l a c e d i n the center of the d i s c s to allow passage of the p e r t u r b i n g metal needle was small enough to be i g n o r e d . The measurement procedure i s d e s c r i b e d i n s e c t i o n 2.2.5 and the experimental r e s u l t s are shown i n F i g u r e 34a. The crosses are f o r the s t r u c t u r e with the s l o t s r o t a t e d and the dots f o r the s t r u c t u r e w i t h s l o t s i n l i n e . The top set of d i s p e r s i o n curves i n F i g u r e 34a give a comparison between the experimental r e s u l t s and both the one and 1 two—mode t h e o r e t i c a l r e s u l t s which i n c l u d e the e f f e c t of the 17 i r r o t a t i o n a l mode. F o l l o w i n g A l l e n and Kino*s analysis, A and C i n equation (3.25) were found to be A = 1.44 C - 0.20 o I f the s l o t s are r o t a t e d by 180 from d i s c to d i s c , C 16 17 1 becomes small and i s assumed to be zero t i o n ^ d i s p e r s i o n curve ( l ) was produced ' « With t h i s assump- t h e o r e t i c a l l y u s i n g the TMQ^Q c a v i t y mode p l u s the i r r o t a t i o n a l magnetic f i e l d . p e r s i o n curve Dis- (2) was computed assuming the presence of the TMQ^Q and T M ^ Q normal modes p l u s the i r r o t a t i o n a l mode. Neither of these t h e o r e t i c a l curves compares a t a l l w e l l with the measured curves although t h e i r shapes are the same. However, we do note 100 that the experimental s t r u c t u r e with s l o t s i n l i n e has a smaller bandwidth than the one w i t h r o t a t e d s l o t s . Thus i t appears the bandwidth d i m i n i s h i n g e f f e c t of the T M ^ n that mode i s g r e a t e r than the expansion e f f e c t of the i r r o t a t i o n a l magnetic mode. Although the t h e o r e t i c a l d i s p e r s i o n curves d i f f e r from the experimental curves by an amplitude f a c t o r , the theory d i d p r e d i c t the observed passband d i m i n u t i o n . I t was found t h a t i f A ' = A and B = 0.081 (B» = 0.10), the bottom s e t of d i s p e r s i o n curves i n F i g u r e 34a were o b t a i n e d . Although the two-mode d i s p e r s i o n curve does n o t correspond to the measured curve, the dominant mode d i s p e r s i o n curve agrees quite well. I f the s l o t s were p l a c e d some d i s t a n c e from the c a v i t y w a l l , the i n v e r s e cosio« t r a n s f o r m a t i o n y i e l d s a c h a r a c t e r i s t i c capacitance which corresponds to B = 0,10, However, the s l o t s were a g a i n s t the outside w a l l i n the experiment. suggest t h a t , i n the c a v i t y passband, these r e s u l t s the i r r o t a t i o n a l mode does not a p p r e c i a b l y change A and that the e f f e c t i v e s l o t capacitance i s c o n s i d e r a b l y l e s s than that p r e d i c t e d by t h e o r y . the changes r e q u i r e d t o produce Perhaps the correspondence t h i s l a t t e r change i s r e q u i r e d because These were noted'above. the s l o t capaci- tance was determined assuming the s l o t to be s t r a i g h t whereas, it i s a c t u a l l y curved. Another suppositions. c a v i t y was b u i l t to provide a t e s t of these I t c o n s i s t e d of f i v e s e c t i o n s and had the same dimensions as the f i r s t c a v i t y except $ = 39.3° ( A = 1.05). This v a l u e of A was chosen because the shape o f the d i s p e r s i o n 101 z 1.2- a = b 1 .81 = A' = 1.50 c — ' 1 .490 "1 1 c A = i . . 4A .20 = c 2 1.1 1: 2: E 010 f v. ^010 ( 1.0 -± A := IRR ( A + E .0 = A ') + [RR ^ A ' 1, - i]i lim i i 0.2 Figure 34a. 0.4 0.6 0.8 1.0 Experimental and T h e o r e t i c a l D i s p e r s i o n Curves f o r a Slot-Coupled Slow Wave S t r u c t u r e 102 curve i s s e n s i t i v e to A near A = 1.0 ' the measurements are shown i n Pigure 34b„. . The r e s u l t s of The p o i n t s .for the s t r u c t u r e w i t h s l o t s i n l i n e are given by the dots and f o r the rotated s l o t structure (180°) by the c r o s s e s . The s o l i d i n the upper graph i s that produced by the one-mode for curve approximation = 0.504 and A * = A = 1.05. We note a/b = 1.81, Z /rj Q that i t does not agree w i t h the experimental r e s u l t s as w e l l as does the corresponding curve i n Pigure 34a. However, we a l s o note t h a t the mode s e p a r a t i o n at the \JJL = 0 p o i n t i s considerably increased. A f t e r computing a number of one-mode curves f o r s l i g h t l y d i f f e r e n t values of ZQ/TJ and A , i t was found that the p a r a - meters ZQ/7] = 0.504, a/b = 1.81 and A ' = A = 1.0 y i e l d e d a curve which agrees v e r y w e l l w i t h the experimental p o i n t s . In f a c t , the agreement i s b e t t e r than was obtained i n the previous measurement and provides experimental support of the zero—mode confluence p r e d i c t i o n * However, i n t h i s case, a small change i n A from the a c t u a l was r e q u i r e d to produce the agreement* I t i s not s u r p r i s i n g that a small s h i f t i n A due to the i r r o t a t i o n a l magnetic A= 1.5 because f i e l d i s apparent near A = 1.0 but not at the shape of the d i s p e r s i o n curve i s i n s e n s i t i v e to A near A = 1.5» The e f f e c t of the i r r o t a t i o n a l f i e l d on B i s d i f f i c u l t magnetic to a s c e r t a i n from the above r e s u l t s * As noted above, the l a r g e change i n B r e q u i r e d to produce ment w i t h experiment agree- suggests that the i n v e r s e cosine t r a n s - formation does not y i e l d an accurate estimate of l i n e capacitance when the s l o t i s p l a c e d a g a i n s t the outside w a l l . A change i n B due to the i r r o t a t i o n a l mode can a l s o be expected. This F i g u r e 34b* Experimental and T h e o r e t i c a l D i s p e r s i o n Curves f o r a Confluent Slot-Coupled Slow- Wave S t r u c t u r e 104 change may be a f u n c t i o n of i n the r o t a t e d s l o t case which could then e x p l a i n the discrepancy between theory and measure- - TC. ment near The smaller passband found f o r the non- r o t a t e d s l o t s t r u c t u r e would then be due to the T M ^ Q normal mode. The p r o p e r t i e s of the s t r u c t u r e due to the i r r o t a t i o n a l magnetic f i e l d w i l l not be considered f u r t h e r here. We see that the e x t e n s i o n of A l l e n and Kino's theory to i n c l u d e the TM-^Q normal mode does not y i e l d a s a t i s f a c t o r y dispersion relation. result. There are two p o s s i b l e reasons for this The normal mode expansion may not be u n i f o r m l y con- vergent or the p o t e n t i a l f u n c t i o n , "V\ , obtained by assuming two w a l l c u r r e n t s may be l e s s accurate than the one obtained by assuming one c u r r e n t . Comparison of a number of d i s p e r s i o n r e l a t i o n s produced from a v a r i a t i o n a l equation given by A l l e n 17 and Kino i n d i c a t e s t h a t non-uniform convergence of the normal mode expansion i s the cause of the t r o u b l e . The argument i s presented below. The v a r i a t i o n a l equation i s obtained i n the f o l l o w i n g way: Consider the t r a n s m i s s i o n l i n e equation j k Z0 y ~ ' h . (J'« . + /_ , n , i n , i 0 V d N n and replace h n » 1 - b y i t s e q u i v a l e n t (equation 3.10). P .).u n, i ' r Since 105 j " • .u n,l r = + . i ' . .u - one may w r i t e — n,i r J 7 - — + 6a0 J k^V. 1 a$ \ 3« i ~.92Z „..(l/ 1 ± . c o s l , «. ^J LI) T\ \ \ n . i 1 0 da0 r f/ */ .i i3"n , '.u i r 1 ( 2 . . .2j„ Y u h B T n n I f t h i s equation i s m u l t i p l i e d by V. and i n t e g r a t e d along the s l o t , the f o l l o w i n g equation r e s u l t s : ^ af j Y J * da0 - ~ / da0 -a^ -a^ a(I> » Z I 2 0 (l + a(F cos^L) / V. * J " . .u" / l n,i r J da0 ^ -a^ ( « l-aj_ n 2 - (0 2 n 7 ) ¥ n (4.1) This i s the v a r i a t i o n a l equation. V where Sv". (<|J) = i = i,0 V I f the p o t e n t i a l f u n c t i o n i s + Sv.(-(ji) = 0 and V. n i s the s o l u t i o n of (4.1), then © = tt + S» 2 12 2 where ft)' corresponds t o the s o l u t i o n V. . n that O© 2 i s of second order i n Sv.. I t can be shown 106 Let the t r i a l p o t e n t i a l f u n c t i o n be that given by (3*18) when the d r i v i n g terms correspond to the TMQ-^Q and TM^Q modes; namely, V = 'A ( l -» i - ^ ) ' + »n ( ^ cos ka(5 cos® c o s k t t c o s c o s cos ka© where 1 = 'C _ Ca cos <E = i 2 2 i k a•— 1 •2 2 and If A •A this t r i a l function i s substituted i n t o equation (4.1) w i t h only two terms i n the s e r i e s on the r i g h t hand s i d e , the f o l l o w i n g v a r i a t i o n a l equation,is . 2k . A * A ( I ) _ 'CC (IT) _ 2k produced: ('A'C sin ^L/2 2 Q , , * A 2 p p urcLa (fl> - a ) A 'OQH) cos (j) a cos 16 Z c +. 'A ( I ) 2 + •cc'cni) 2 p cos f 1 PA'C* + T A* cXi)(lD , cos 32 k H ix 2 a 2 ft> Z sin l/^L/2 2 2 Lc (o> 2 Q - « 2 3 2 2 ' )(l - k a ) QA«C* + 1 A . » ) A 2 + « a 4k a cos <JJ 2 'A*'c)(ni)(H) 2 ka cos | where ( m 2 2 (4,2) 107 (I) = (ka^ - tan ka(p) (n) = (2 ka cos (ji tan ka$ - ( k a 2 (LU) = (cos 2 2 + l ) cos (p s i n $ - (k 2 2 a f tan ka<j) - ka s i n $) The d i s p e r s i o n r e l a t i o n i s obtained by t a k i n g v a r i a t i o n s i n t h i s expression w i t h respect c o e f f i c i e n t s of the p o t e n t i a l f u n c t i o n . terms i n the V a r i a t i o n a l expression, By i g n o r i n g c e r t a i n two d i s p e r s i o n r e l a t i o n s corresponding to the f o l l o w i n g c o n d i t i o n s a) I f only the f i r s t the f i r s t to the amplitude can be derived: term i n the s e r i e s of (4.1) and term ('A) of the t r i a l f u n c t i o n are r e t a i n e d , the r e s u l t a n t d i s p e r s i o n r e l a t i o n i s b) I f two terms i n the s e r i e s of (4«l) but only the first term of the t r i a l f u n c t i o n are r e t a i n e d , the resultant dispersion relation i s ! 2p n 4 + (P ^ 2 4 2 (fi 2 i)(m) - l) (ft 2 2 - 2 (io /« ) )(l) 2 3 2 1 (4.3) -l)$) 108 A program was w r i t t e n to produce s o l u t i o n s of d i s p e r s i o n r e l a t i o n (4,3). The s o l u t i o n s were found to be i d e n t i c a l with those given by the two-mode d i s p e r s i o n r e l a t i o n . can be explained This r e s u l t by the f a c t that the two d i s p e r s i o n r e l a t i o n s are i d e n t i c a l i f 2 p ft(m) 2 = _ 1 (D(n) Numerical t e s t s show that t h i s r e l a t i o n i s almost, i f not e x a c t l y , s a t i s f i e d i n the range of v a r i a b l e s of i n t e r e s t . From the d i s c u s s i o n above, i t can be seen that the use of the two—term p o t e n t i a l f u n c t i o n d i d not by i t s e l f y i e l d the inaccurate the dispersion relations. I t i s the second term i n s e r i e s on the r i g h t hand side of equation ( 4 . l ) which i s responsible f o r the e r r o r . The e r r o r r e s u l t s even when the one—term p o t e n t i a l f u n c t i o n i s used. of equation (3.10) f o r h Through s u b s t i t u t i o n i n t o the t r a n s m i s s i o n Q line equation, the behaviour of the s e r i e s elements were made dependent upon the convergence p r o p e r t i e s of the normal mode expansion. In p a r t i c u l a r , the c o r r e c t i o n f a c t o r i n equation (4.3) i s dependent upon the r a t i o p l o t t e d i n Figure 35. (£1 2 - l)/(.T2 2 2 - (w^/fl)^) ) which i s In f a c t , f o r two v a l u e s of A , the 109 n- i to. ^ 2 ft" 2 F i g u r e 35. A P l o t of ( H - l)/(ft - (©y^) ) f o l l o w i n g expressions were found to hold! A = 1.2 ^ Q. 1.5: 2 p ^ 4 (p a 2 A = (fl* ( m ) ^ 4 - n (i) (ft 2 2 2 - 1) 1.71 -(^) J 2 2 2 - 1) (ft - (wyV ) 2 1.0: 2 p 4 n (P a 2 4 2 ( m ^ -1) - i) (D (n 2 2 2 (D. 2 = 1 < 8 5 (n -&) ) i 2 Thus the nature of the a d d i t i o n a l depend p r i m a r i l y on the r a t i o (\f2 was (ft 2 2 - l) - (« /« ) 3 1 2 f a c t o r appears to - 1)/(<Q 2 2 - (<a^/n^) ) which obtained from the general theory of normal mode expansions, 110 I t i s p r e c i s e l y t h i s a d d i t i o n a l f a c t o r which causes the d i s p e r s i o n r e l a t i o n t o i n a c c u r a t e l y p r e d i c t the d i s p e r s i v e p r o p e r t i e s structure* I t might be argued that the constants, would be much smaller 1.71 and 1.85, and an accurate d i s p e r s i o n r e l a t i o n would r e s u l t i f a more accurate p o t e n t i a l f u n c t i o n were used. argument ignores of the But t h i s the f a c t that the one-4term t r i a l p o t e n t i a l f u n c - t i o n y i e l d e d very good d i s p e r s i o n curves when only one normal mode was used* B e s i d e s , the v a r i a t i o n a l equation i s r e l a t i v e l y i n - s e n s i t i v e to e r r o r s i n V\» pansion i s not uniformly TM^^Q I t appears t h a t the normal mode ex- convergent when ordered such that the^ mode i s the second term. There appears to be no n a t u r a l way of o r d e r i n g the normal modes i n the c a v i t y expansion except by resonant frequency. A number of arrangements have been i n v e s t i g a t e d i n the t h e s i s (TM 0 1 0 -TM 1 1 0 , ™oiO~™011' ™010~™110~ s l o t rotated) s and were not found to give b e t t e r r e s u l t s than given by the dominant mode dispersion r e l a t i o n . The ^ Q l O ' ^ l l O P * a r ^ s o r < iered to ascending resonant frequency;, i n the cases s t u d i e d . according Of the d i s p e r s i o n r e l a t i o n s i n v e s t i g a t e d herein,, the dominant mode d i s p e r s i o n r e l a t i o n i s the most u s e f u l . 111 5. A CONFLUENCE CONDITION FOR THE CENTIPEDE AND RELATED STRUCTURES 18 The centipede reversed-loop-coupled structure i s a l o g i c a l extension of the 5 19 cavity chain ' . The l a t t e r has s e v e r a l loop p a i r s per s e c t i o n and the former has loop p a i r s f i l l i n g e n t i r e c a v i t y circumference* I t has been noted i n Chapters the 1 and 3 t h a t it-mode confluence i n each of these s t r u c t u r e s should 5 be p o s s i b l e . The work of Dunn, Sabel and Thompson , who used a lumped c i r c u i t approach i n t h e i r a n a l y s i s of a loop-coupled c a v i t y chain,was d i s c u s s e d i n the i n t r o d u c t i o n . The work of 19 Pearce i s also relevant* From F i g u r e 7 of h i s paper, we see t h a t when the stopband between the loop and c a v i t y passbands i s made s m a l l , the d i s p e r s i o n curves do not have a broad maxima or minima at = TC* In other words, behaviour c h a r a c t e r i s t i c of a n e a r - c o n f l u e n t s t r u c t u r e has been observed* An a n a l y s i s based Upon f i e l d theory has been performed by Bevensee f o r the reversed-loop-coupled v a r i a t i o n a l procedure relation: [| P + | 2 P l 2 c (l ( l c a v i t y chain. He used and obtained the f o l l o w i n g d i s p e r s i o n + cos^L) + \ p - cos^L) - k 2 2 c (l - cos^L) - k - m (P 1 2 c - k )(P 2 2 1 | P 2 (l 2 x - k )cos 2 2 1 + cosU, ^ = 0 (5*1) where P = normalized frequency of the s h o r t - c i r c u i t cavity c mode (E x n = 0 p c and H; h = 0 on = normalized frequency bdy.) of the o p e n - c i r c u i t c a v i t y mode ~ 112 E x n = 0 E" • n = 0 (== : _ _ on the w a l l s , - r r , „ on the c o u p l i n g H • n = 0 H x n = 0 L & planes) P, = normalized frequency of the s h o r t - c i r c u i t loop E x n = 0 mode (TT ^ on c a v i t y midplane and on metal H • n = 0 J r surfaces) p^ = normalized frequency E • n = 0 ( TT :a_i _ H x n = 0 of the o p e n - c i r c u i t loop mode . E x n = 0 c a v i t y midplane and rr. :_ • „ on metal H • n = 0 J r surfaces) and = cross-coupling c o e f f i c i e n t l m I t i s not necessary to give a more d e t a i l e d of the constants above except resonant curves (constant)' explanation to note t h a t the normalized f r e q u e n c i e s mark the e x t r e m i t i e s of the d i s p e r s i o n (see Figure 36). The d e t a i l s are a v a i l a b l e i n Bevensee's 26 book . In p r a c t i c e , the constants above are determined experimentally. P , P-^ and p c are measured d i r e c t l y and p^ and c m^ are determined i n d i r e c t l y by f o r c i n g the d i s p e r s i o n r e l a t i o n to f i t experimental TI/2 points). r e s u l t s at two other p o i n t s ( i . e . , the two I t has been observed that the d i s p e r s i o n r e l a t i o n 26 then agrees w e l l with the other experimental points provided there are l e s s than e i g h t loop p a i r s . The behaviour of the centipede s t r u c t u r e , which u s u a l l y has eight or more loop p a i r s , i s q u i t e w e l l d e s c r i b e d by the 26 f o l l o w i n g e m p i r i c a l d i s p e r s i o n r e l a t i o n given by Bevensee : | P ( l + cosl^fL) + | P d - cosVpL) - k | P ( l + cosl/. 2 2 c | 2 P l 2 c (l - cosl/^L) - k 2 - K sin 2 x 2 \pL = 0 (5.2) 113 Pigure 36, D i s p e r s i o n Curves f o r a Loop-Coupled C a v i t y Chain Ve note t h a t t h i s equation d i f f e r s only s l i g h t l y equation (5.1). from Again the constants are determined e x p e r i - mentally and Bevensee r e p o r t s very good agreement with measurement. I t w i l l now be shown t h a t p = p^ i s a necessary c and s u f f i c i e n t c o n d i t i o n f o r p r e d i c t i o n of Tt-mode confluence by equations simple. (5»l) and (5.2). The proof f o r n e c e s s i t y i s q u i t e If dispersion relations (5.1) and (5.2) are d i f f e r e n t i a t e d w i t h r e s p e c t to ^ L , w i t h .^pL subsequently to rc(k = P c or p-^), the f o l l o w i n g equation r e s u l t s : / (P Ve set 2 c C " Pn 1 2\ dk ) —7- dlpL = A 0 see t h a t a non-zero group v e l o c i t y at the Tt-mode can only occur i f p = p, . c ^1 The proof f o r s u f f i c i e n c y i s more i n v o l v e d and w i l l r be g i v e n f o r the reversed-loop—coupled r e s u l t f o r the centipede Dispersion r e l a t i o n structure. The f i n a l s t r u c t u r e w i l l then be stated,, (5»l) can be w r i t t e n [A - k ] [ B - k j - ^ ( 1 + c o s \ / ; L ) ( P - k ) ( P 2 2 2 2 c 2 x - k ) = 0 2 where A = \ P (l 2 c + coslpL) + | p 2 (l--\cosl/fli) and B = \ P If 2 1 ( l + cosl^L) + | P 2 x ( l - cos-l//L) t h i s equation i s expanded and regrouped, the f o l l o w i n g 2 q u a d r a t i c equation i n k jl - + coslpL)] k results: - |A + B - ^ | ( P 4 + AB - + P 2 C (1 + coslpL) P Now i f the confluence c o n d i t i o n , p 2 C a )(l P 1 2 + cosl/^L) k = Q 2 = p , i s enforced, the 2 solution for k is J2 - m ( l + cosl/jL)j k | (P + P 2 1 2 c 2 1 = P )(l 2 d ( l - cos + c o s l / ; L ) ( l - n^) ± R (5,3) where R = \(1 + c o s l / ; L ) 2 l/« 2 I(P 2 c r, 2 x 2 - 2 P l m.P P 1 c 1 ~ 2 ) 2 .. l + COSXJJL) T _ 2 a 2x2/ l 2 m ) 2 ^ - 2 (20^- (1 + cosl/i)) ] 2 2 m1(l / c + ^ ( 1 - coslpL)sin \/;L [ p + 2, m + ^|(P + P i P 2 c P 2 n 4 c j-^ -P 2 C (P 2 C + P )] 2 x 1) 2 115 D i f f e r e n t i a t i o n of the l e f t hand side of equation (5»3) = 71 and k ='p with subsequent a p p l i c a t i o n of the c o n d i t i o n s yields , 4 Repeating dk. # L c P t h i s procedure on the f i r s t side y i e l d s zero. Thus dk A 4 P term on the r i g h t hand ^ ,. d^TL c = ± l x dR dlpL m Now i(P 2 c - P *)* —2( P c + P 1) ' 2 2 - m. P \ 1 c 1 + — 2 - 2 2 1 i - - 2 ( + coa\ph) T + 2 [2 coslpL'(l - cosl/^L) + s i n l / ; L ] [ p 2 - P (P p, )] 2 2 C + C 2 4 c -am^V} ¥hen\j(/L = 71, k = p , then c { ) = - 2 » 1 ( J Since the d e r i v a t i v e of 2 C - I 2 c ) ( P C 2 - P 1 2 » j with r e s p e c t to i s f i n i t e , we can write lim fr - -i<p. -* ><p -V> " » 2 2 e 2 c ^ 116 But sin^L lim T R = lim . 1 2 cosl/^L + nu m ^ sinU/T, - m 2 P 1 + ^ (1 V 1 c m. 1 3 Pc 2 m (l + c o s \ p h ) 2 2 C P 2 1 sxn Now 1 + COSIJOL s m o i = XJJL] i COSI/;L) (i+ m x/jL-^ sin and 1 lim XJJL. Tt + cos^L _ s i n Thus lim l/^L Tt sinl/jL _ R V P 2 0 - * ><p - P i ) 2 c c 2 - • P > c * * l ) 4 P 1 lim P s i n ^ L L cos^L) - COS ^ D 0 + 2 c 2 3 2 P 1 2 - 1 Consequently, . dk 2 ^c V ra1 (p - „ P' 1 7 v 2< , c = + g JM (V - cL ± C P 2 2 C )(P 2 C - P! ) 2 4P, c i s negative, positive. ^ 4p . or Since 2 P *)(p c *c n x± d \ph v 2w I f the the expression under the root s i g n i s same procedure i s a p p l i e d to equation ( 5 . 2 ) , the r e s u l t i s Thus we see t h a t Bevensee s two 1 p r e d i c t confluence at the -rc-mode i f f p = c the n-mode group v e l o c i t y f o r the proportional through the was y C C C confluent zero-mode i s p r o p o r t i o n a l I t i s to be expected that K i n ( 5 . 5 ) i s a l s o r e l a t e d to the bandwidths of the Since t h e o r e t i c a l d e t e r m i n a t i o n of the frequencies p c and p^ i s very d i f f i c u l t , one the dimensions of a confluent The to centipede s t r u c t u r e may e x c i t e d i n an n - s e c t i o n c a v i t y midplanes. two passbands. resonant would determine How~ prove to be very u s e f u l i n c a v i t y Tt-mode can c a v i t y by p l a c i n g e l e c t r i c The expression structure experimentally. a c c e l e r a t o r a p p l i c a t i o n s since the the structure. i t w i l l be remembered that B i s the main bandwidth- determining f a c t o r . ever, the passbands S i m i l a r behaviour -L circumferential-slot-coupled confluent also note that reversed-loop s t r u c t u r e i s I 2 2 2 2 ' \/(p - P ) (p - P, ) . factor found f o r the and p^. ' We to the bandwidths of the two group v e l o c i t y at the /IT d i s p e r s i o n r e l a t i o n s do be shorts at s t r u c t u r e i s c e r t a i n l y worthy of further investigation. In t h i s case the overlapping c a v i t y t e s t discussed i n the i n t r o d u c t i o n does not appear.to apply. s h o r t - c i r c u i t modes,whereas here we c i r c u i t modes. The It i s valid for are concerned with open- c a v i t y o p e n - c i r c u i t mode i s e x c i t e d i n the c a v i t y formed by p l a c i n g magnetic s h o r t i n g planes over coupling holes. the The loop o p e n - c i r c u i t mode i s e x c i t e d i n a c a v i t y c o n s i s t i n g of the volume enclosed by two s h o r t i n g surfaces at the c a v i t y midplanes. magnetic In t h i s case, the dual of the overlapping c a v i t y c o n d i t i o n given e a r l i e r a p p l i e s ; i.e., i f the two each of the two c a v i t i e s formed by p l a c i n g magnetic shorts at sets of symmetry planes have equal f r e q u e n c i e s , the corresponding fluent. resonant slow wave s t r u c t u r e i s con- Although t h i s t e s t does not lend i t s e l f to p h y s i c a l / a p p l i c a t i o n , i t and the previous one form a symmetric pair.;. „ U s u a l l y i n standing wave p a t t e r n s , nodes of the t r a n s verse e l e c t r i c f i e l d are separated by nodes of the magnetic f i e l d . For t h i s reason, transverse one might say that the t e s t s d i s c u s s e d above are not independent. two In other words, i f one t e s t can be a p p l i e d to a s t r u c t u r e , the other can a l s o be applied. Although t h i s statement i s o f t e n t r u e , i t i s not always so* For example, i f one of the zero-mode standing wave p a t t e r n s i n a c o u p l e d - c a v i t y system i s given by the c a v i t y TMQ^Q mode, there i s no set of appropriate symmetry where a magnetic short can be placed without fields. any Thus although s t r u c t u r e , both may The two for one not of the two planes p e r t u r b i n g the t e s t s i s r e l e v a n t to be. t e s t s d i s c u s s e d above have been shown to be it-mode confluence. But what of zero-mode confluence? valid Con- •119 s i d e r the c i r c u m f e r e n t i a l — s l o t - c o u p l e d c a v i t y c h a i n . mode of the c a v i t y passband corresponds The zero- to the c a v i t y s h o r t - c i r c u i t mode (TMQ^Q) but the zero-mode of the s l o t passband corresponds to the s l o t o p e n — c i r c u i t mode c a v i t y midplanes). ( mE = 0 at Here ve compare an e l e c t r i c a l l y shorted c a v i t y w i t h a m a g n e t i c a l l y shorted one. The symmetry of the s i t u a t i o n suggests t h a t confluence t e s t s i n v o l v i n g c a v i t i e s with the same t e r m i n a t i o n are v a l i d f o r it-mode confluence and t h a t those i n v o l v i n g c a v i t i e s with d i f f e r e n t t e r m i n a t i o n s are v a l i d f o r zero-mode confluence* 6, AUTO-CONFLUENCE In t h i s chapter i t w i l l be shown t h a t s t r u c t u r e s are not, i n general, However, before c o n s i d e r i n g auto-confluent useful accelerating structures. the arguments l e a d i n g to t h i s 7 conclusion, the reasoning used by Lewis i n discovering this c l a s s of s t r u c t u r e s w i l l be reviewed. In Chapter 1, i t was noted that the nodes of the standing wave p a t t e r n s at the . n^- and -n^-modes are at planes of symmetry and t h a t , as a r e s u l t , the frequency and f i e l d s of these modes i n the d i e l e c t r i c - l o a d e d s t r u c t u r e correspond to the resonant frequency and f i e l d s of the c a v i t i e s shown i n Figure the 4. I t was f u r t h e r noted t h a t i f these two c a v i t i e s have same resonant frequency, the i n f i n i t e p e r i o d i c corresponding to them i s c o n f l u e n t . structure Lewis proposed that i f one could produce a p e r i o d i c s t r u c t u r e f o r which the two corresponding Tc-mode c a v i t i e s were i d e n t i c a l , then the s t r u c t u r e must be automatically structure confluent. shown i n Figure He then introduced 37. the This s t r u c t u r e even-offset s a t i s f i e s the i d e n t i c a l Tt-mode c a v i t i e s c o n d i t i o n as one can see by comparing the c a v i t y terminated by planes a-a with t h a t termimated by planes b-b. In a n a l y z i n g the s t r u c t u r e , Lewis obtained d i s p e r s i o n curve by assuming t h a t the lower gap E X shown i s (-E e"^ Q V ^ ) . By u s i n g (+ E e~^ P 2 a wave w i t h opposite Q one component obtains However, the d i s p e r s i v e behaviour of s t r u c t u r e i s completely d e s c r i b e d by one d i s p e r s i o n r e l a t i o n since the r e s t of the diagram can be f i l l e d symmetry p r o p e r t i e s . 2 group v e l o c i t y and, as a consequence, the dashed d i s p e r s i o n curve. the field P ^ ), V the s o l i d i n by use of '••121 b r— 1 i ' y • ! 1 z 1 a Pigure 37. An Even-Offset Curve b a S t r u c t u r e and I t s D i s p e r s i o n D i s p e r s i o n curves computed by Lewis do show non-zero group v e l o c i t y at the TC—mode; i . e . , confluence. However, l e t us consider the s t r u c t u r e d u s e f u l n e s s i n a c c e l e r a t o r a p p l i cations. In h i s a n a l y s i s , Lewis obtains y 122 co E q\ E (y,z) = — / sin p p Q z q/2 m sinh S ( b y ) - ( - l ) m sinh S (b-y) m + q / 2 m s i n h 2S b m m At the Tt-mode, the e l e c t r o n beam couples to the m = 0 space harmonic of the forward wave and the m = -1 of the backward wave. On the a x i s (y = 0), the these space harmonics are p r o p o r t i o n a l sinh S b - sinh S b = 0 m m m = 1: sinh S b + sinh S b = 2 sinh S b m m m I t can be f i e l d s are i d e n t i c a l . Lewisi only holds on the The of the space harmonics i s shown t h a t t h i s r e s u l t holds independent of the f u n c t i o n a l form of the gap gap amplitudes:of to m = 0: In other words, the amplitude of one zero. space harmonic f i e l d provided top and bottom This r e s u l t , which was observed by axis. i m p l i c a t i o n s of t h i s r e s u l t on the a x i a l beam c o u p l i n g p r o p e r t i e s of auto—confluent very important and c y l i n d r i c a l structures are c l a r i f i e d i n a d i s c u s s i o n by Crepeau 27 and Mclsaac structures. on the consequences of symmetry i n p e r i o d i c The p e r t i n e n t p a r t s of t h e i r work are presented below. C e n t r a l to t h e i r d i s c u s s i o n i s the i n t r o d u c t i o n of field operators corresponding to v a r i o u s and d e s c r i b i n g the r e s u l t i n g f i e l d the t r a n s l a t i o n operator structure symmetries. d e f i n e d by For " T E ( r , 0 , z ) = E ( r , 0 , z + L) z z ~ symmetries instance, are 123 and corresponding to a p h y s i c a l s t r u c t u r e symmetry d e s c r i b e d by F(r,0,z + L) = F(r,0,z) i s introduced. The Floquet c r i t e r i o n allows us to w r i t e TE (r,0,z) -= e - ' ^ z J which i s an eigenvalue equation. E(r,0,z) L Although t h i s r e s u l t i s not novel, the technique has been f r u i t f u l l y a p p l i e d to s t r u c t u r e s of more complicated Let symmetry* us determine the symmetry p r o p e r t i e s r e s p o n s i b l e f o r the auto-confluent property of the e v e n - o f f s e t s t r u c t u r e . Notice t h a t i f the top h a l f of the s t r u c t u r e i s r o t a t e d 180° about y = 0 and then s h i f t e d by p/2 i n the z d i r e c t i o n , i t then c o i n c i d e s with the lower h a l f * I t i s t h i s symmetry property which i s r e s p o n s i b l e f o r auto-confluence. The corresponding symmetry i n c y l i n d r i c a l s t r u c t u r e s may be d e s c r i b e d by F(r,0 + TI, z + |) = F(r,0,z) and i s a s p e c i f i c example of what Crepeau and Mclsaac screw symmetry. The f i e l d operator i s d e f i n e d by S E (r,0,z) = E (r,0 z call z + TX, z + |) (6*l) Again an eigenvalue equation a r i s e s ; i . e . , S E (r,0,z) = s E (r,0,z) (6*2) The eigenvalue must c o n t a i n the f a c t o r e ^ ^ ^ - 2 to s a t i s f y the Floquet c r i t e r i o n and hence can be w r i t t e n . = .S e - ^ / L 2 (6.3) The E z expansion f o r a c y l i n d r i c a l slow wave s t r u c t u r e may be expressed as a double F o u r i e r s e r i e s j i . e . , oo oo n m where r m ' Now by combination of ( 6 . l ) - (6.4) oo oo S E (r,0,z) = \ z ' ' / n -i ) / Z m E i (r) e ^ < ^ > -°° _o=» e" n znm oo -i 7 v oo J P m 1 znm 7 n m _ oo - oo ' For an eigenvalue t o e x i s t , we must have P = e - j (m+n)7x or, i n other words, the sum (n + m) must always be e i t h e r even or odd. Thus w r i t e m + n = 2V + a or n := •— m + 2V + a where a = 1 or 0 and V i s an i n t e g e r such that -co^y^oo Although there are two p o s s i b l e values of a, there may not be two p h y s i c a l l y e x c i t a b l e modes i n p r a c t i c e . event, consider Table 2. 9 In any the two p o s s i b i l i t i e s which are summarized i n As u s u a l , we are i n t e r e s t e d i n the m = 0 and m = -1 125 space harmonics at the c o n f l u e n t Tt-mode. Note that f o r e i t h e r value of a, n i s odd f o r one harmonic and even f o r the other* a == 1 a == 0 m = 0 m = -1 m = 0 n is his ri i s Table 2 » : h is even odd odd even m = -1 , Consequences of Screw Symmetry T h i s f a c t i s very important f o r the f o l l o w i n g reason. amplitude f o r s t r u c t u r e s that have an empty factors, E z n m (r), The r e g i o n around the symmetry a x i s (r = 0) vary e i t h e r as I ( T^*") or as J (K r ) where I i s the m o d i f i e d Bessel f u n c t i o n of order n m ' n A n. These f u n c t i o n s have, a non-zero value f o r r = 0 only i f n = 0. Thus i n the beam i n t e r a c t i o n r e g i o n (r ~ 0 ) , there i s no a p p r e c i a b l e i n c r e a s e i n shunt impedance at the Tt-mode of c y l i n d r i c a l auto-confluent s t r u c t u r e s . There i s another h e u r i s t i c argument which appears to l e a d to a c l a s s of auto-confluent s t r u c t u r e s d i s t i n c t from t h a t d i s c u s s e d above. doubly p e r i o d i c Consider Pigure 38. s t r u c t u r e shown i s (to , a l t e r n a t e d w i t h fl* ,') w i t h p e r i o d i c l e n g t h 1 SX 2L» The SX I f \p^"L i s the phase change over L, then, i n g e n e r a l , = Tt/2 ; but as fl>^ i s made equal to there i s a stopband at © , , t h i s stopband must disappear. Thus i t appears t h a t we SX have a c o n f l u e n t s t r u c t u r e . V I I t h a t the amplitudes However, i t i s shown i p Appendix of the space harmonics a s s o c i a t e d with the dashed curve are z e r o . Thus the dashed d i s p e r s i o n curve 126 si (A Cavity ai. Passbands ft ft "•sl ^ " s l ft ft, —J— n/2 %/2 Figure 38. A Doubly P e r i o d i c Slot-Coupled C y l i n d r i c a l S t r u c t u r e w i t h T y p i c a l C a v i t y Passband D i s p e r s i o n Curves cannot be s a i d to e x i s t * I t i s a l s o argued i n the appendix that t h i s r e s u l t i s general. 7. The 127 CONCLUSIONS uneven-offset s t r u c t u r e has been analyzed i n such a way t h a t the two sets of gap f i e l d s are not a p r i o r i The increased equal. accuracy of the d i s p e r s i o n r e l a t i o n d e r i v e d herein i n comparison with Lewis' d i s p e r s i o n r e l a t i o n i s due p r i m a r i l y to a proper matching of the f i e l d components a t the gap mouth although the a d d i t i o n of an evanescent s l o t mode produced a s l i g h t a d d i t i o n a l improvement. The a n a l y s i s corroborates Lewis* p r e d i c t i o n of c o n d i t i o n a l it-mode confluence f o r t h i s structure. The experimental r e s u l t s i n d i c a t e that the dimensions of a confluent uneven-offset s t r u c t u r e are not a c c u r a t e l y p r e d i c t e d by the d i s p e r s i o n r e l a t i o n d e r i v e d herein. The exact dimensions are probably best determined e x p e r i m e n t a l l y . The a n a l y s i s by A l l e n and Kino of the c y l i n d r i c a l slot- coupled c a v i t y chain has been presented and extended to account f o r more o f the normal modes. I t was shown t h a t tt , =tt..i s a si 1 necessary and s u f f i c i e n t c o n d i t i o n f o r p r e d i c t i o n of c o n d i t i o n a l zero—mode confluence between the f i r s t c a v i t y and s l o t passbands by A l l e n and Kino's dominant mode d i s p e r s i o n r e l a t i o n . I t was a l s o shown that t h i s d i s p e r s i o n r e l a t i o n does not p r e d i c t it—mode confluence between the c a v i t y passband and the second s l o t passband i f tt~L< 2' 63 s ^ n e p r e d i c t i o n of zero—mode confluence was confirmed e x p e r i m e n t a l l y . Two approximate analyses of the c y l i n d r i c a l c a y i t y chain with l o s s were performed. t h a t the presence of l o s s i n a confluent any slot-coupled The r e s u l t s i n d i c a t e s t r u c t u r e does not make s i g n i f i c a n t change i n the mode s e p a r a t i o n i n a 10 or 20- section cavity. They also i n d i c a t e that the group v e l o c i t y at the confluent zero-mode, i s dependent: upon;the r e l a t i v e values of 128 the Q of the c a v i t y short c i r c u i t mode and the Q of the s l o t open c i r c u i t mode. The d i s p e r s i o n r e l a t i o n obtained by r e t a i n i n g the TMQ^Q and T M ^ ^ Q normal modes i n the c a v i t y expansion y i e l d e d curves which d i f f e r g r e a t l y from those produced by the dominant mode dispersion relation. I t was found that these two-mode d i s - p e r s i o n curves do not agree with experimentally curves. determined A v a r i a t i o n a l a n a l y s i s i n d i c a t e d that the inaccurate r e s u l t s are due to a non-uniform convergence of the normal mode expansion i n the order used. In the absence of any o ; : v method of o r d e r i n g the normal mode expansion other than that of ascending resonant frequency, the dominant mode d i s p e r s i o n r e l a t i o n appears to be the most u s e f u l * I t was shown that the d i s p e r s i o n r e l a t i o n s d e r i v e d by Bevensee f o r the reversed-loop-coupled c a v i t y chain and the centipede s t r u c t u r e p r e d i c t c o n d i t i o n a l ix—mode confluence. d i s p e r s i o n r e l a t i o n f o r the reversed—loop s t r u c t u r e was d e r i v e d from a v a r i a t i o n a l equation which i s based upon f i e l d r a t h e r than an equivalent circuit. The theory The c o n d i t i o n f o r ix—mode confluence i s both necessary and s u f f i c i e n t . An o p e n - c i r c u i t ix-mode confluence t e s t , which i s complementary to the s h o r t - c i r c u i t ix-mode t e s t , was presented. In a d d i t i o n , a "mixed" t e s t (one o p e n - c i r c u i t c a v i t y and one s h o r t - c i r c u i t c a v i t y ) was presented f o r zero—mode confluence. A d i s c u s s i o n of auto-confluence based upon an a n a l y s i s by Crepeau and Mclsaac showed that auto—confluent structures do not g e n e r a l l y have an enhanced Z , (resonant) a t the ix-mode. 129 APPENDIX I TRANSMISSION LINE TO WAVEGUIDE TRANSFORMATION FOR THE OFFSET STRUCTURE The uneven-offset t r a n s m i s s i o n l i n e can be made i n t o a waveguide by p l a c i n g s h o r t i n g planes at x = 0 and x = a* s l o t and l i n e E z The equations must be r e w r i t t e n i n t h i s event since must be zero at the side w a l l s . The new expressions now be given f o r the case i n which there will i s one v a r i a t i o n i n the x direction. The f i r s t s l o t mode, which i s TEM i n the t r a n s m i s s i o n l i n e , "becomes a i T E ^ mode i n the waveguide and i s d e s c r i b e d by A a) upper slot An TC H y ~ E H z ka77 _ j = A y- cos sin x -Wf x = " sip S] where E = H = 0 x z and B^= b) lower E sin 3 - 8 (1^ _ y ) - y) > (1-1) i n 2 | o s B ( L - y) C x TC\2 (2L) a slot = A j 0 e " ^ A B^ 2 k7/ p / 2 s i n Sf- s i n B ( L TCX 0 + y) s i n — cos (3( 2 a L + (1-2) •;'' 130 The second s l o t mode i s a: TM mode i n the t r a n s m i s s i o n l i n e and, thus, i s a i s that more f i e l d field y E (k . upper s l o t 2 + r l S Ei B l = c o iji s i n | ( z+ (z + § ) s i n h J 2 T s = (% (k z ) i n 2 | cos & b) E 2 s i n vhe.re y 2 -I - l = " ZaTTK E In s h o r t , the }c o g h y^^ _ y ) 7 (—) = B z components are now p r e s e n t . The d i f f e r e n c e components are a) = TM mode i n the waveguide* z + !> + (J) - k 2 2 c o 2 s h ^ 1 - y) 2 lower s l o t + "/' )B e " ^ 2 3 2 s i n S — y T & f — — = B , 8 ^ sin B£ <s < (L - y) 2 a C Q S 22L r ( z r cos y (L +y) - i n ^ ( . - _ E=£) J ( ^ s i n h h y, ^ + y) z ~^ 2 2 131 jrkB H - e 2 cos m 2 *r)7 s i n 21 ( z _ ( E t * ) ) o s h 7' ( L . +y) 2 "2 C v 2 -jkB e 2 H s i n S£- o s ^ - where C 7' (z - ( ^ ) ) cosh 7 « ( L + y) 2 v 2 = ( ^ ) + (*) - k 2 2 2 Note that both E (1-4) and H z f i e l d components are present i n z the second s l o t mode as they are i n the other evanescent modes. 2 Thus a h y b r i d wave i s r e q u i r e d i n the l i n e . slot This wave w i l l be c o n s t r u c t e d from a TM and TE mode with the r e l a t i v e amplitudes being determined through the matching p r o c e s s . the TM mode P oo E s i n h £ (b + y) -^m TCX = sin - H = x f m oo + P' s i n h $ (b - y) m 'm P cosh $ m oo sin ^ m m -jB z m a/ P sinh £ m 'm Q m _ oo -P« cosh $ (b - y) m *m ' v oo = (b + y) m dr - oo H z x J (b + y ) ' J 2 m +P' s i n h g (b - y) m nr ' J -jB z First 132 P OO E x TC = a -JP 6_ TCX cos — a m — oo = j sin +P> s i n h $ (b - y) P cosh % (b + y) 'm m J m 0 8 TCX z S _i OO E s i n h g" (b + y) a m m -P' cosh % (b - y) _ oo where £ 2 m m v (71)2 2 v a' (1-5) and f i n a l l y the TE mode H = k fj Q *m cosh 0 (b + y) nm • ' J -js TCX Jt — C 0 S m -Q' cosh 0 (b - y) *m m • . — oo v 8 = ^ 77 x 17 y £ T J kix a77 • S i n m Ttx \ "7/ 7 +0' Q nn «2 m _ oo z s i n h £> (b - y) cosh ^ ( b + y) m m -j3z m jS m -JPm Jt oo = x cos EL a m H 'm J s i n h f& (b + y) nil Q *m oo H v z m -G m cosh ^ ( b - y) 133 Q E x m +Q y -i 2 = nk J % . — a/ m J -2 S m TXX \ sm — a s i n h ^ ( b - y) m Q cosh # (b + y) *m 'm ' oo E J m = j k cos — m - oo „ s i n h # (b + y) 'm ' > m 0 -Q* cosh 0 (b - y) *m 'm ' m — oo v J (1-6) The E f i e l d component z i s matched i n e x a c t l y the same manner as before and the f o l l o w i n g equations are obtained: P m = 2 sin g m A q/2 x sin 3 l p sinh 2 ^ b 'm r p 1 m sinh m (—) - P q 2 *m (1-7) and P 1 m = 2(-l) m s i n B r/2 m A 2 s i n P p s i n h 2$ b X 2 B 2 m P m x s i n h ^2 : r' 1 r m (1-8) Note that i f k 2 i s changed to B 2 = k 2 - (it/a) 2 i n equations (2.12) and.(2.15), the above equations r e s u l t . I t w i l l be seen at the end of t h i s d i s c u s s i o n that the above t r a n s f o r m a t i o n i s general with the consequence that the normalized transmission l i n e frequency, k^, i s r e l a t e d to the normalized waveguide frequency, k^, f o r an otherwise i d e n t i c a l s t r u c t u r e by 134 kw = \k V t + (*) a' 2 d-4) v Since there are now more amplitude constants to r e l a t e , more f i e l d and at H components must be matched. components, the E x non-gap s u r f a c e . © 0 a m P + k m m 2 z z f i e l d component w i l l be matched x the upper and lower s l o t s . same as that f o r the E In a d d i t i o n to the E The matching procedure i s the component since both are known on the The r e s u l t s of t h i s match are 2 j* Q m *m S m a^ 2( l B S i n %h h S p sinh 2 0 b [ ( ^ L ) nil ^ q' i Pm n ^ 2 - 8 1 m J 2 2 r r (1-10) and («' B ™ p m a m m + k 2 *Lm m 2(-l)- S =- 2 m (J) B sinhT l 2 2 sin J 2 r/2 ~m m p sinh 2 ^ b 'm (12L) v r 7 _ fl "m 2 (1-11) F i n a l l y the H x field component i s matched as before The r e s u l t s f o r the- upper s l o t are (0 P + S. 8 Q ) cosh 2$ b ^m m a m *nr *m v 1 2k' 8 q cos B 1^ sin 8 m r m -(# m v, P' + ^ 0 m a m r Q*) *m q/2 u / S 8 in m 2 -oo and r (1-12) 13 5 47, 1 - (0 oo q cosh T ^ ' ^ l ' P + - 8 Q ) bosh 20 b 8 m -(g P ' + - p Q*) m m • a m *nr — oo sin0 m I f equations (1-10) and ( i - l l ) are solved f o r and are then s u b s t i t u t e d with equations (1-7) and (1-8) i n t o (1-12) and (1-13), the f o l l o w i n g two homogeneous equations result: 0 = A, "4p " coth 20 b s i n 8 q/2 _m *m 1 P m sin 8 1 ^ (-1) " s i n p 1 0 8 *nr m i 2 — oo oo n sinh T ^ ! / 1 q/2 s i n P m 2 m r/2 sinh 2 # b m v c o t h 2# b s i n ~ 8 2 m -oo 2 B^ m m 2 2 2 ~~ m k + % B S m m — oo oo + A, PL and equations - sin 8 1 . (1-13) R e s u l t s f o r the bottom s l o t are s i m i l a r . pq cos 81^ (—) -3 ' m v, *Lm (-l) sinh 7* 1 2 2 2 m 'm (i2L) _ q m q/2 m _ B ' m sin p (—) m m q/2 s i n p - p 2 m r/2 s i n h 28:b m (1-14) and oo 0 = A n coth 20 b s i n p q/2 m m s i n pi.^ m —oo q/2 M 136 oo sin -A, 31 \ - l ) sin B m q/2 s i n m r/2 2 0. m m q' v s i n h 2$ b m m K — oo k B, 2 pq cosh T^-^i 8 m 2 r sinh T^ ! 1 8 % m 2 r m -oo coth 2$ b s i n 8 m (—) . - q/2 ^ 3 ' q m oo k B, '(-l)" 2 s i n h yn. " 2 2. 8 1 2 sin 8 q/2 s i n 8 m r/2 m A s i n h 2$ b m m r- OO (1-15)- The other two equations are s i m i l a r . d i f f e r e n c e between equations I t can now be seen t h a t the only (2.20) and (2.22) and (1-14) and (1-15)- i s that k i s r e p l a c e d by 8 = Wk 2 and B n n . In the determinantal change i s r e q u i r e d . o r i g i n a l determinantal - (it/a) 2 i s r e p l a c e d by k B /8 only the f i r s t equations relation relation, Thus the s o l u t i o n of the (eqn. 2.25) i s k f o r the t r a n s - m i s s i o n l i n e and 8 f o r the waveguide. Hence the f o l l o w i n g relation is valid: k w =^/k 2 t + U/a) ' 2 Note t h a t t h i s t r a n s f o r m a t i o n r e l a t e s the t r a n s m i s s i o n line s o l u t i o n to the waveguide s o l u t i o n with one v a r i a t i o n i n the x direction. 137 APPENDIX I I UNEVEN OFFSET THEORY FOR k > B The symbol S ffl = j $ m w i l l be u s e f u l i n t h i s because the e f f e c t of k becoming greater 2 the s i g n of S • m Although k > P m than 0 m discussion i s to change only occurs f o r m small i n the f i r s t few passbands, the theory to f o l l o w w i l l be f o r k a l l m. P for m I t w i l l become apparent that the i n d i c a t e d changes only apply to s e v e r a l elements i n p r a c t i c e . (2.2), Consider equation •JP ~m z J E z P s i n h S (b + y) + P' s i n h S (b - y) m m ' m m ' = J J m OO -JPm z Jr P s i n h j $ (b + y.) + P» s i n h j $ (b - y) m — CO oo P s i n $ (b +J y) + P" s i n $ (b - y) m *m ' m ^m* ' = J z -JP J m —oO Similarly H = - P k m cos $ (b + y) - P Mn J ' T m cos $ (b - y) T m m m oo Thus P s i n h 2S b m m P' s i n h 2S b m m becomes J j P m s i n 2$ b ^m j P" s i n 2$ b m ^m J J -iP Jr m z 138 J P cos 2$ b - P' m m m m P cosh 2S b - P' m m m m and P - P m m 1 cosh 2S b m P - P' cos 2$ b m m m m m Now i t can be seen t h a t equations (2.12) - (2.17) become j 2 sin B m q/2 m B , sink l . 1 1 1 A, 8 m p s i n 2$ b r T K m 2 - K 2 2 2 oo P cos 2$ b - P' m m m j2k q cos k l ^ •1 sinh yLl-. ' 2. 1 r sin p T T T m m q/2 nr m m -oo 8 (P B, q cosh -- T^l m j2 ( - l ) s i n 8 m A 2 = cos 2$ b - P») m m 3 ~ $ (K - 8 ) m 2 m ' 2 v -c© P' = m m JS JS B r/2 0 2 sin 8 m sink l p s i n 2$ b m s i n h y»l 2^2 2 2 K' - 8 ^ 2 ^m A 8 2 2 m P.* cos 2$ b - P m m m 3 $ 'mm .j2k r cos k l ^ r m q/2 K (-l) m m s i n 3 r/2 m — oo B 2 - 43 oo yx (-l) 8 m r co sh / 2n 2, m (P m cos 2$ b - P M s i n 8 __m m r/2 $ (K' - 3 ) Mil 2 *m ' 2 m m 2 _ oo I t f o l l o w s from these r e l a t i o n s t h a t equation (2.20) becomes 139 0 qp cos k i j = JL 4k q / 2 cot 2$ b sin 8 2 + sin kl. TP -i *mm r m oo - A. (-l) sin k l . m j m $B mm oo 1 B, sin sin 8 m q/2 s i n 2$ b m 2 m — oo - sin 8 r/2 m B q / 2 cot 2$ b m ^ m 2 T sinh T^ ! 1 oo + B, sinh (-l) 7n 2 2 " i $ (K 7 m sin8 m 2 2 - 8 2 m m r/2 sin 8 ) s i n 2$ b m q/2 m m — oo The other equations are changed i n a s i m i l a r manner. the r e s u l t above shows t h a t the f i n a l equations In f a c t , set of homogeneous undergo the f o l l o w i n g t r a n s f o r m a t i o n s i n the elements of the <T s f o r which k > T coth 2S b m m 8 : cot 2$ m 1 -1 b m 140 APPENDIX I I I EVALUATION OF NORMAL MODE EXPANSION COEFFICIENTS Maxwell's equations V ^ n , i f o r the normal modes are = -J"n i * V i f V * \,i = °n,i n , i V ^ ; , i = 3 « n i V- n , i j< £ ¥ (HI-1) f f where n. x E . = 0 1 n, 1 _ on i ' S. + S! + SV i i n. . H . = 0 l n, l and Maxwell's equations f o r the a c t u a l f i e l d s , l T ( p ) and ^ ( p ) , are V x E ± V x H ± = -j« u = ja e E H ± ± (HI-2) V x l j * = j a [i HJ V x H* = - j a e E* where n. x E. = 0 I l on S^ only "n. . H. = 0 I Now consider I V . (E, x H* . ) dv v• 1 n ,• i. V x E.1 dv - H* // E.i . V x H* n , i. dv v v v. 1 = / u H..H* . dv + i» / l n,l n, l - jo* J r // J v. v. l l e E.. E* . n,i I V i t h the use of Gauss' theorem, the above r e l a t i o n may be written to / e E. . E* . dv - / n,i » n,i I / v. v. l i 4 / (E. x H* ii / R H.. I H*' . dv n , iI . ) • n• ds S.+ S.'+ S*' i i = - i i / / d v . ) . n- ds n,i' i (E. x H* l S! + S»' i l since n^ x E^ = 0 on S^. V . (E£ >;L x H.) yields tt n,i S i m i l a r l y , c o n s i d e r a t i o n of u H., I * . dv - / r l n,i (o / e E. . E* . dv / l n,i 142 (E* . x H.). n. ds n, 1 1' l = -J %+ s£ s[+ (III-4) H..(n.x E* .) ds n, i ' S i + S! + sv 1 0 Equations (III-3) and (III-4) are l i n e a r i n the volume and may be solved to give f~ 10) jtt n, / . J. _ ( IE , x H * . ) . n. _ . ds ( l n,i' l s.*+ sv e E. . E* . dv = I n, I — i_ 2 2 to - tt . ' v n l jtt _ u H. . H* . dv = I n, I R I integrals / (E.xH* s>+ 's».» D : —— 1 I . ) , I . ds n,I' x 2 2 tt - « . l 143 APPENDIX IV ZERO-MODE GROUP VELOCITY WITH THE IRROTATIONAL MODE 1 6 Allen's dominant mode d i s p e r s i o n r e l a t i o n which i n c l u d e s the e f f e c t of the i r r o t a t i o n a l mode i s 2B» sin ^L/2 = 2 ^ "^ ) tan A» H u / 2 - A ' Sl%/2 l where B' = B \jk - C cos ^ L A> C 7 tt'j = a> j A - C cosl/^L sl and, as a r e s u l t A' = 1 <fi \j A - C cosl/^L ' sl For confluence to occur, the e f f e c t i v e s l o t resonant for \jj = 0 must be s e t equal to tt^. "l = * s l \l A " Thus A' - \lk - C cosl/^L ' 1 C " frequency 144 where A' = - r ^ A-C and C = C A-C Henee the c o n f l u e n t d i s p e r s i o n r e l a t i o n i s 2 B ! . 2 s iLnn na -n ) 2 xbh -TF-— • = fi TI/2 tan \JA'- C* ( l i t / 2 \ZA*- C COSI//L" cos^L' and i f i t i s d i f f e r e n t i a t e d with respect to A(,Q,) + B ( f i ) ± L Jc(ft) = , the r e s u l t i s H ( l - f i * ) C L sinl/jL 2(A - C cosl/;L) [ D(ft) - C(ft)] C 2 ^(l-,n )(sec 0 ! _ ( ) 2 2 - 1) 2B'(tan 0 - 0 ) ( A ' - C »cos \ph) / 2 where (V-l) A ( f t ) = 1 - 3 £l d -niu 1 -n )(«ec 2 (tan 0 - 0)( fid D(ft) and 3 2 \j A'- -n ); 2 2B'(tan 0 - 0) 0 = A' ' f i n / 2 Now A(l) = -2 C(l) 0 B(l) = ? D(l) oo o -1) C cos IpL ) 2 145 Also -ft )CL sin 2 as ft — 1 0 2(A - C c o s l ^ L ) . and C'(Tc/2)ft (l -ft )(sec Q 2 2 2 2B' ( t a n 0 - 0 ) ( A ' - C cosl^L) 2 C'(Ti/2)ft (l 2 - f t 2 - cos Q) 2 as F i n a l l y , by use of lim ft— ft — Jj2 cosl/^L) 1 ' l ' H o p i t a l ' s r u l e , i t can be shown that C ( f t ) D ( f t ) =. 8 B VA - C TC 1 if is lim 3 / / 2 2 ) ^ ! 2 B ' ( s i n 0 - e cos © ) ( A ' - C» — 0 - 1) finite B ( f t ) = -2 ft — 1 From these r e s u l t s and ( V - l ) , * i dTTJ = ± «i L B \/A^C' 2TC A P P F I N D I X V A N A L Y S E O F A LOSSY CYLINDRICAL SLOT-COUPLED STRUCTURE 1, An A n a l y s i s Assuming Lossy S l o t s Consider the s l o t t r a n s m i s s i o n l i n e to have s e r i e s re- sistance 57$ " = 5T0 ( r + j w L = ^ . i + ) Assume r « fl)L. One then 6 v h - ^ c y i obtains 2 i 2 % + k'^ V . = -k'Zj" A h, • where *; - d - isr) *o - k = z . " Proceeding i n the same way as i n Chapter 3* one obtains 2B s i n ^'L/2 = 2 m n l = l ) ©'-tan©' where ©' = ft'A f = CQA §) y By reducing _ r _ To ^ - 2wL - n = o (i-j?) the d i s p e r s i o n r e l a t i o n to f u n c t i o n s of and ocL o n l y , one obtains | sin lpL = - ( E + F 2 2 2 - E) - and y(E + F 2 2 - E) 2 + F 2 (V-l) s i n h aL ; = 2F —:—rhr s i n WL 147 where to(ft -l) 2 E [G(QG-C)+H(QH+D)] 2B[(OG-C) +(©H+D) ] 2 2 (V-2) -n(n -l)(HC-fDG) F = 2 2B [ ( © G - C ) + ( © H + D ) ] 2 +71 G = A H = I 2 -7A A = cos © cosh T'© I sin 0 sinh = C = sin 0 cosh'X© D = cos © sinh7© 2. An A n a l y s i s Using a Lumped E q u i v a l e n t Circuit A l l e n ^ k has shown t h a t the simplest e q u i v a l e n t circuit which approximates the s l o t - c o u p l e d system i s an i n f i n i t e of the c i r c u i t «o ^ = l / \J^ i^ i g s s chain shown i n F i g u r e 39 where »^ = l / /V^C]"* a n d ^i/^ i = s 2BA/TC.» The s e r i e s resonant c i r - 2C, L /2 1 - n n n n — | 'si si F i g u r e 39» E q u i v a l e n t C i r c u i t cuit 'si of the Slot-Coupled System represents: the c a v i t y and: the: shunt: resonance c i r c u i t repre-r sents the s l o t . :.•.•••.: V. The e f f e c t of c i r c u i t ..• l o s s can be i n v e s t i g a t e d by p l a c i n g a s e r i e s r e s i s t a n c e , r ^ / 2 , i n each of the s e r i e s arms and a shunt 148 conductance, & i> s ^ "^ n ne shunt arm. By use of the A B C D p a r a - meters and the F l o que. t c r i t e r i o n , i t can be shown that . 2 ij^'L *—-— = — 1 — —l s iinn 4 Z Z sl where \p* = \jj + joe . By reducing t h i s d i s p e r s i o n r e l a t i o n to f u n c t i o n s of and ctL o n l y , one obtains ( V - l ) where TC E = 8B AH' (ft 2 - D(ft 2 - V?) - A n ' MsiA 2 (v-3) ,-TC 8B A f t (ft 2 - i) 2 iA s ( f t - i/A ) 2 + 2 Qi and = the Q of the s e r i e s resonant c i r c u i t Q ^= g the Q of the shunt resonant c i r c u i t , APPENDIX V I THE FLOQUET AMPLITUDES Consider the c y l i n d r i c a l s t r u c t u r e i n which the dominant mode i s being e x c i t e d e l J 0 ( K l = 0<V> ) m J r ) a m e m Thus a, e ~ J • 2mn L = e^ e m m m I f the t h i c k n e s s of the end w a l l s i s £~ , then 2^2 e^ e a L = m m m , dz or 2e *m 2e 1 FT r At the zero-mode m B m L s i n —x— l£+ ^ c o s —V^ — s .i n "h£ —TT* cos — and Thus a l a 0 _ s i n TT <^/L TT [ l + from which the f o l l o w i n g t a b l e i s computed? i &> 0.03 0.06 0,09 0.12 0.15 0.18 Table V - l . a i / / a 0 0.029 0.057 0.081 0.104 0.125 0.144 Floquet Amplitude Ratio 151 APPENDIX VI i; REDUCTION TO SINGLY PERIODIC STRUCTURES R e f e r r i n g to Figure 38, consider the f o l l o w i n g F o u r i e r a n a l y s i s of the E f i e l d component i n the s t r u c t u r e shown: z E E n -jn0 (r) e znm ' -3 m B z m e E (r,0,z) k z k 0 < z < e E zk; L € z « 2L k (r,0.z-L)e where In the usual way, one obtains E m znm (r) e-J* ' n0 n = Fm (e.) k + ( - l') Fm (e») k' m v where or :."F (e, ) m k' e m k / e 0 m E ( r , 0 , z ) dz z k 152 But when « , = S .L , e, = e' and a SX J£ d i s p e r s i o n curve i n Figure of m's, J£ in =0 f o r m odd. The dashed 38, which corresponds to the odd set thus cannot be s a i d to e x i s t . There i s a good p h y s i c a l reason to expect t h i s r e s u l t to be g e n e r a l . Assume i t i s not. Then, f o r some s t r u c t u r e , reduction of a doubly p e r i o d i c s t r u c t u r e to a s i n g l y p e r i o d i c one y i e l d s two d i s p e r s i o n curves (the s o l i d one and the dashed the same argument, i f a t r i p l y p e r i o d i c s t r u c t u r e one). By ( i . e , <p ^» s .40', and (p", a l t e r n a t i v e l y ) were reduced to a s i n g l y p e r i o d i c one, three d i s p e r s i o n curves would r e s u l t . th of an n S i m i l a r l y , reduction r l y p e r i o d i c s t r u c t u r e would y i e l d n d i s p e r s i o n c u r v e s . As n approaches i n f i n i t y , the n d i s p e r s i o n curves would merge to form a d i s p e r s i o n s t r i p by f i l l i n g the area - c o ^Y^L ^ o o , Since t h i s r e s u l t i s not observed i n nature, the assumption above must be invalid. 153 REFERENCES 1. Slater, C , Microwave E l e c t r o n i c s , Van Nostrand, York, London & Toronto, 1950, New 2. F r y , D. V. and Walkinshaw, V., "Linear A c c e l e r a t o r s " , Reports on Progress i n P h y s i c s , v o l . 12, 1948-49, p. 102. 3. S l a t e r , J . C., "Design of L i n e a r A c c e l e r a t o r s " , Review of Modern P h y s i c s , v o l * 20, 1948, p. 473. 4. Walker, G„ B. and West, N* D., "Mode Separation at the it-Mode i n a D i e l e c t r i c Loaded Waveguide C a v i t y " , Proc. I . E J . . p a r t C, v o l , 104, 1957, p. 381. 5. Dunn, P. D., Sabel, C, S, and Thompson, D. J., "Coupling of Resonant C a v i t i e s by Resonant Coupling D e v i c e s " , Atomic Energy Research Establishment Report GP/R 1966^ 1956. 6. West, N. D., i n v e s t i g a t i o n of D i e l e c t r i c Loading i n a L i n e a r A c c e l e r a t o r , Ph.D. Thesis, U n i v e r s i t y of London, 1958. 7. Lewis, E. L., Confluent Pass Band C h a r a c t e r i s t i c s i n High Frequency Delay L i n e s , Ph.D. Thesis, U n i v e r s i t y of London,1961. " 8. Walkinshaw, W. and B e l l , J.j'S.,, "Review of Theory of Metal Loaded L i n e a r A c c e l e r a t o r " , Atomic Energy Research Establishment, Report G/R 675. 1951. 9. Murdoch, D. C., L i n e a r Algebra f o r Undergraduates, John Wiley & Sons, New York, 1957. p 10. Grosjean, C. C , "On the Theory of C i r c u l a r l y Symmetric TM Waves i n I n f i n i t e I r i s l o a d e d Guides", II Nuo,vo Cimento, v o l . 1, 1955, pp. 427-438. 11. Vanhuyse, V. J . , "On the ((3Q,K) Diagrams for Circularly Symmetric Waves i n I n f i n i t e I r i s l o a d e d Waveguides", II Nuovo Cimento, v o l , 1, 1955, pp. 447-452. 12. L i n e s , A. W., N i c o l l , G* R., and Woodward, A. M., "Some P r o p e r t i e s of Waveguides with P e r i o d i c Structures ', P r o c . I JS.E. , v o l . 97, Part 3, 1950, p. 263. 1 13. H e f f n e r , H., " T r a v e l l i n g Wave A m p l i f i c a t i o n of M i l l i m e t e r Waves", T e c h n i c a l Report No. 51, E l e c t r o n i c s Research Laboratory, Stanford U n i v e r s i t y , C a l i f o r n i a , 1952. 14. Ramo, S. and Whinnery, J . R., F i e l d s and Waves i n Modern Radio, John Wiley and Sons, New York, 2 Ed., 1953, P.. 317. n a 154 15. Courant, R. and H i l b e r t , D., Methods of Mathematical P h y s i c s , v o l . 1, New York and London, I n t e r s c i e n c e P u b l i s h e r s , 1953. 16. A l l e n , M. A., "Coupling of M u l t i p l e - C a v i t y Systems", M.L. Report No. 584, Microwave Laboratory, ¥.¥. Hansen L a b o r a t o r i e s of P h y s i c s , S t a n f o r d U n i v e r s i t y , S t a n f o r d , California. 17. A l l e n , M. A. and Kino, G. S., "On the Theory of S t r o n g l y Coupled C a v i t y Chains", T r a n s a c t i o n s of IRE, PGMTT. v o l . 8, no. 3, (May 196077~ 18. Chodorow, M.. and C r a i g , R. A., "Some New C i r c u i t s f o r High-Power Traveling-Wave Tubes", Proc. IRE, v o l . 45, pp. 1106-1118, Aug. 1957. 19. Pearce, A. P., "A S t r u c t u r e , Using Resonant Coupling Elements, S u i t a b l e f o r a High-Power Travelling-Wave Tube", Proc. IEE (supplement No. l l ) , v o l . 105, p a r t B, pp. 719-726, Dec. 1958. 20. Teichmann, T.. and Wigner, E.P., "Electromagnetic F i e l d Expansions i n Loss-Free C a v i t i e s E x c i t e d through Holes", J o u r n a l of A p p l i e d P h y s i c s , v o l . 24, no. 3, (March 1953). ~~ 21. Kurokawa, K., "The Expansion of Electromagnetic F i e l d s i n C a v i t i e s " , T r a n s a c t i o n s of IRE, PGMTT. v o l . 6, no. 2, ( A p r i l , 195877" " 22. Ramo, S.. and Whinnery, J . R., op. • c i t . 14, p. 194. 23. Chodorow, M., Pearce, A. F., and Winslow, D. K. , "The Centipede High-Power Traveling-Wave Tube", ML Report No. 695, Microwave Laboratory, W.W. Hansen L a b o r a t o r i e s of P h y s i c s , S t a n f o r d U n i v e r s i t y , May 1960. 24. Ramo, S. 25. Bevensee, R. Mi, Electromagnetic Slow Wave Systems, John Wiley and Sons, New York, 1964, p. 215. 26. I b i d , j Chapter: V I I . 27. Crepeau, P. J . and Mclsaac, P. R., "Consequences of Symmetry i n P e r i o d i c S t r u c t u r e s " , Proc. IEEE, v o l . 52, no. 1, pp. 33-43. and Whinnery, J . R., Op. c i t . 14, p. 135. Ginzt.on, E.L., Microwave Measurements, McGraw-Hill, New York, 1957, Chapter 10.
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Study of confluence in periodic slow wave structures McDiarmid, Donald Ralph 1965
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