UBC Theses and Dissertations

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UBC Theses and Dissertations

Study of confluence in periodic slow wave structures McDiarmid, Donald Ralph 1965

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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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