BROOKHAVEN NATIONAL LABORATORY Associated Universities, Inc.

Upton, L.I., N.Y.

THE DEFLECTING MODE I N THE CIRCULAR IRIS-LOADED WAVEGUIDE

OF A rr

H. Hahn

October 25, 1962

. ,

I ~ . , :. I L - , I. Introduction . >

, r ' 1' - - L , .

t The r f pa r t i c l e separator, proposed i n 1959 by W.K.H. Panofsky [l: and now -

* - - * - ,,in preparation for the Brookhaven Alternating Gradient Synchrotron, requires

, .

a r f s t ructure which gives a transverse impulse t o a passing r e l a t i v i s t i c par t ic le . - 1 I

7 I n order t o produce an accumulative transverse deflection of a traveling

-;charged pa r t i c l e with an electromagnetic f i e ld , it is necessary tha t the f i e l d

~ o n t a i n s a synchronous component and i n principle , waveguides and cavi t ies a r e

equivalent with respect t o the pa r t i c l e dynamics. It was pointed out by '

. a -

12' , H.G. Hereward r2j , t ha t the e l e c t r i c and magnetic deflection of a transverse : gt.

4- C

r 1' 2 s t e l e c t r i c mode (i.e., with no e l e c t r i c f i e l d component pa ra l l e l t o the direct ion - ?

-T-lr.~$of the pa r t i c l e velocity) cancel exactly a t a l l pa r t i c l e veloci t ies . The ' '

deflecting force of a transverse magnetic mode on a synchronous pa r t i c l e w i t 2

the velocity v is proportional t o the factor 1 - (v/c) and vanishes

fore i n the case of r e l a t i v i s t i c pbrticles.

It i e impossible t o achieve a synchronoue d e f l e c t h n , i n an ordin

Nevertheless, it is poesi5le t o def lect par t ic les by a TM - mode. Again, a

TE - mode cannot be used [3-j . W.K.H. Panofsky proposed a c i rcu lar cyl indrical

cavity h i c h is fed a t two points t o exci te a c i rcu lar ly-polar i~ed TMlll mode

4 . P.R. Phi l l ips used a TMOIZ mode in a rectangular cavity and va

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-2- HH- 5

obtain a particle deflection K5- i - . P. Lapostolle shored, that a TMl10 - mode in a circular cavity can give equally a deflection r6 1. In both cavities (TMOl2, TMllO) the fields have no dependence in direction of the traveling

particles. The phase velocity is therefore infinite and the limiting factor

2 1 - (v/c) does not appear. Houever, the deflection is now proportional to

the transit time factor sin(~~/2), where 7 is the transit time and.^ the

rf frequency. For nonsynchronous deflection it is therefore impossible to

increase the maximum transverse impulse in a single cavity by a greater length.

The use of a multiple-cavity structure is however possible.

Deflecting electromagnetic modes, which are neither TE or TM and structures 7 -'

tq support them were indicated by J.P. Blewett i7,87 and H.G. Hereward 1 2 1 . - i - E a - -

A proper choice of the dimensions a312ows a* phase velocity equal or close to

l@t velocity and a synchronous deflection is possible, Some of the proposed

rf structures have the disadvantage of vanishing group velocity, which makes

the excitation of the wanted mode critical. An extensive study of deflecting

modes in rectangular coordinates was made by R. Gabillard [ 9 1 - . The interpretation of the beam blow-up (pulse shortening) in

electron linear accelerators requires the existence of a backward-wave in which

TE and TM modes arc combined in such a say as to leave a residual transverse

deflection on a synchronous relativistic particle 110 1 . The usefulness of this field configuration as a deflecting mode was recognized at Stanford [16] . For the time being, an iris-loaded waveguide seems to be the best deflecting

structure available and it will be used for the projected rf separator in

Brookhaven.

The aim of this report is to show an analysis of the deflecting mode in

an iris-loaded waveguide which is not based on the concept of.TE and TM modes

a& generating Eields. The hybrid solutions employed are derived from the tranverse

-3- HH- 5

components of the electric and magnetic Hertzian vectors. Their use avoids

the singularities in the field expressions for phase velocities equal to the

velocity of light, which must occur with TE and TM modes. The fundamental

properties of the hybrid solutions are exposed. The lowest mode of the hybrid .

group is the wanted deflecting mode. The exact relationship between frequency,

phase velocity and geometry of the structure can be derived for the deflecting

mode in the form of an infinite determinant. A simple approximate so.lution,

which is sufficient for a mode identification is indicated.

11. The Hertzian Vectors

The electromagnetic fields in vacuo may be written in terms of the electric ' 2 3

and magnetic Hertzian vectors, Ile and , as follows:.

where the Hertzian potentials are solutions of the vector Helmholtz equation

The time dependence is always assumed to be ejWt and the equations are

written in the MKSA unit system. The wave number k and the characteristic

Impedance Z0 have the usual meaning

-4- HH- 5

The most general solut ion of t he vector Helmholtz equation requires th ree A A 2

independent vector solut ions P , Q , L . The solut ions can always be brought

i n a form so t ha t

The most general solut ion f o r t he Hertzian vectors i s therefore a l inear A 2 1

combination of P and Q . It i s convenient, i f P has a 1ongi.tudinal component 2

only and Q transverse,components only. Vector solut ions i n t h i s spec ia l form

were derived fo r c i r cu l a r cylinder coordinates ( r , 8, z ) i n reference 111 :

- 5- HH- 5

2 2

The a l t e rna t i ve solut ion Q' i s i r re levan t for our with k = $ -k y

problem, as the f i e l d s derived give no t ransverse def lect ion.

It is a well known f ac t , t ha t the most general representation of an

electromagnetic f i e l d requires only two independent vector solut ions 62:; . The two vector solut ions may be obtained from the two Hertzian vectors with

longitudinal components only. It is equally poss ible t o use instead the two

Hertzian vectors with transverse components only. In the f i r s t case, the

f i e l d s a r e developed i n terms of TE and TM modes; i n the second case, the f i e l d s

a r e developed i n terms of two hybrid solut ions . The choice of the generating

solut ions w i l l be made i n each individual case t o y ie ld the most convenient

expressions.

111. Transverse and Hybrid S o l u t i o ~

The expressions fo r t he e l e c t r i c and magnetic f i e l d components which can d 4

be derived from the two solut ions P and Q fo r the e l e c t r i c and magnetic

Hertzian vectors a r e tabulated i n Table 1. The e l e c t r i c Hertzian vector 3 4 -\ .A

ne = P leads t o TM modes, the magnetic Hertzian vector l$, = j P t o TE modes.

2 a It i s proposed t o c a l l the f i e ld s derived from lIe = jQ as hybrid e l e c t r i c

-a 2%

(HE), t he f i e l d s derived from $ = Q as hybrid magnetic (HM) solut ions .

The hybrid solut ions obtained and used i n t h i s repor t a r e d i f f e r en t from the - symmetric hybrid waves introduced by G. Goubeau [ 1 3 ~ .

The inspection of the f i e l d expressions reveals several i n t e r e s t i ng

fea tures , The 0.- order ( v = 0) hybrid solut ions , HMO and HEo, s p l i t both

i n two independent TMo and TEo solutions. Modes without €)-dependence a r e

therefore always transverse. This statement holds i n c i r c u l a r waveguides w i t h

i r i s loading.

A t cu tof f , i . e . , for 8-40 and y+k, the HM and HE solut ions tend t o become

TM and TE f i e ld s .

-6- HH- 5

The l imi t ing case when the phase v e l o c i t y approaches l i g h t ve loc i ty , i . e . ,

6 - ; k and y + O , is of s p e c i a l i n t e r e s t . TM and TE so lu t ions become both

TEM waves with no longi tudinal f i e l d components. I n p a r t i c u l a r t h e TM1 and

TE so lu t ion approach both a plane wave. On t h e other hand, t h e hybrid so lu t ions 1

have s i x non-vanishing f i n i t e f i e l d components which a r e given i n Table 2 ,

A The Lorentz fo rce on a charged p a r t i c l e with t h e v e l o c i t y v equals

The p a r t i c l e is assumed t o t r a v e l synchronously (v = v ) i n d i r e c t i o n of the ph

wave propagation

v v J k - = - e , = - 0 with $ = 7 C C B z ph

The fo rce components a r e then given by

Using t h e f i e l d expressions of Table 1, i t can be e a s i l y v e r i f i e d t h a t

t h e TE modes exe r t no fo rces on a p a r t i c l e .

The ~naximum t ransverse force exerred by a TM mode i s found t o be

- 7- HH- 5

and'vanishes, as reported, for r e l a t i v i s t i c par t i c les .

The HE f i e l d exerts a transverse force

- Y

A '

FT = 9 1

The HM f i e l d exer ts a transverse force r

The maximum transverse force on a r e l a t i v i s t i c pa r t i c l e , due t o HE and HM f i e l d s ,

does not vanish and i t s value i s - -1

where a and bo a r e the r e l a t i v e amplitudes of HE and HM f i e l d s . 0

The 1. order hybrid modes (v = 1) a re of spec ia l i n t e r e s t , as t h e i r

combination leads t o t he def lect ing mode. Here the transverse force is aberra-

t ion f ree , i , e . , t he force vector i s constant for a l l r and 0 :

The def lect ing mode has therefore on a synchronous, r e l a t i v i s t i c p a r t i c l e t h e same

e f f ec t as a uniform, s t a t i c e l e c t r i c f i e l d of the equivalent f i e l d s t rength

HH- 5

4

The complex Poynting vector S is defined a s

and t h e time average power f l u x densi ty S i n propagation d i r e c t i o n is given by z

where t h e a s t e r i s k s i g n i f i e s t h e complex conjugate value.

I f t h e phase ve loc i ty equals c, SZ of t h e hybrid so lu t ions is found t o be .-+

b L 2

0 k r " kr - - 2v cos 2vB + - - v + l 1 2 2u+lv:-

In con t ras t t o t h e t ransverse modes, t h e power f l u x dens i ty has always

negative values i n v i c i n i t y of t h e z a x i s (r = 0 ) . The hybrid modes a r e there-

fo re inherent ly backward waves, even i f t h e in tegra ted power flow assumes

p o s i t i v e values. The hybrid so lu t ions a r e not orthogonal and t h e cross product aobo

does not vanish.

The t r ansverse and hybrid solut ioris of Maxwell's equations have been found

without considering t h e boundary condit ions, But a so lu t ion has physica l s i g n i f i -

cance only i f a supporting m e t a l l i c s t r u c t u r e can be found. I n c i r c u l a r

c y l i n d r i c a l waveguides, t h e HE and HM so lu t ions do not s a t i s f y t h e boundary

conditions. The only poss ib le modes a r e TE and TM f i e l d s .

-9- HH- 5

It w i l l be shown t h a t i n an i r i s - loaded waveguide a combination of HE

and HM so lu t ions does s a t i s f y t h e boundary conditions, One hybrid so lu t ion

alone cannot e x i s t . The lowest passband hybrid so lu t ion with u = 1 w i l l

be used a s de f lec t ing mode (HEMl1).

I V . F ie ld Expansions for t h e Hybrid Modes

It remains t o prove t h a t hybrid modes do s a t i s f y t h e boundary conditions

i n t h e given s t r u c t u r e and t o der ive t h e r e l a t i o n between frequency, phase

ve loc i ty and guide dimensions. It seems impossible t o f ind '.a so lu t ion f o r the

f i e l d d i s t r i b u t i o n i n closed form. The a l t e r n a t i v e i s t o develop t h e so lu t ion

i n t h e form of an i n f i n i t e s e r i e s . This method was suggested by d.C. Hahn f o r

t h e ana lys i s of cav i ty resonators (14. It was modified by J.S. Be l l and C -

'd. dalkinshaw f o r t h e ca lcu la t ion of t h e acce le ra t ing mode i n i r i s - loaded

waveguides k53.

The ~ e r i o d i c i t y of t h e i r i s - loaded waveguide makes it s u f f i c i e n t t o consider

one c e l l only (Fig, 1). ,The c e l l i s cut i n t o two spaces with simple boundary

condit ions.

Ser ies expansions of t h e f i e l d s a r e found f o r t h e s l o t region (region 11)

arid i n tht: cen te r region (region I ) . ax well'.^ equations show t h a t it is neces - sa ry and s u f f i c i e n t t o match EZ , HZ , E8 , H a t t h e common boundary between

8

region I 'and I I , l ead ing t o an i n f i n i t e s e t of homogenous l i n e a r equations. The

secular determinant of t h i s s e t of equations gives t h e required frequency

condition. Approximate so lu t ions are found by l imi t ing the number of elements

i n t h e underlying matrix.

HH- 5

Fig. 1

1. The F ie lds i n Region I1

The f i e l d s i n region I1 a r e obtained as an i n f i n i t e Fourier s e r i e s of

a l l poss ib le modes, s a t i s f y i n g individual ly t h e boundary condit ions a t

z= f d/2 and r P b. The complete f i e l d expressions requ i re two s e t s

of a r b i t r a r y r e a l c o e f f i c i e n t s (cm, c: and dm, d ) corresponding ' to

TM and TE modes. A l l abbreviat ions used a r e defined i n t h e appendix and

t h e 8 - dependence i s not e x p l i c i t l y ;mitten.. The f i e l d s take then tZhe form

Q) Q

z H I1 = Z d B (rmr) j-l s i n qmz + Z da By (r; t)- cos TI; z 0 2 1 m v I

HH- 5

co vZ (T'r) 'XI . v . rn k + j , { c l - r + d; Ti B~ (r; r) ) C0s 7; 2

1 rm (r; r) rn

. cD

k - j c f c h T z v . ( r ~ r ) + d l m 7 Tm TA r s i n I]; z 1. .

. 2. The'Fields i n Region I

The f i e l d s i n region I a r e obtained as an i n f i n i t e Fourier s e r i e s of

waves propagating i n z - di rec t ion with d i f f e r en t phase ve loc i t i es . The

per iod ic i ty of t he s t ruc tu r e indicates t h e use of Floquet's theorem t o

determine t he possible propagation cQnstants

The individual term need not s a t i s f y by i t s e l f t he boundary conditions

on the i r i s e s a t r = a, and therefore t he f i e l d s may be .developed i n terms

of HE and HM modes. The most general expression requires again two i n f i n i t e

s e t s of a r b i t r a r y r e a l coef f ic ien t s (an, bn). The f i e l d s i n region I may

be wr i t t en a s

V. Matchlr? of t h e Fie lds a t t h e Common Boundary -- '

The four s e t s of c o e f f i c i e n t s r e q c t r e four i n f i n i t e s e t s of eruat ions which

a r e o b t a i ~ e d by matching of fogr f i e l d components a t r = a. A poss ib le choice

is EZ, HZ, Ees He . Maxwell's equations ensure then t h e si inultanews matching ;

of H and Ers what can be seen from t h e following r e l a t i o n s r

1. Maeshieg o f Ez 3f r = a.

EZ I has t o s a t i s f y t h e boundary condit ion on t h e i r i s e s

HH- 5

and t h e .following r e l a t i o n holds

A t t h e common boundary between region I and region I1 we have

I E = EZ Z

I1 and the re fo re

By using t h e or thogonal i ty of t h e Fourier terms, t h e f i r s t s e t of

equations i s obtained i n t h e genera l form

2. Matching of EQ a t r = a.

&analogous procedure applied t o Ee yie lds t h e second s e t of

equations i n t h e genera l form

3. Matching of HZ a t r = a.

The two s e t s of equations obtained by matching EZ and E express e each c o e f f i c i e n t i n region I i n terms of t h e c o e f f i c i e n t s i n region 11.

The matching of HZ and He on t h e o the r hand, can be done i n such a .lay t o

express t h e c o e f f i c i e n t s i n region I1 a s a l i n e a r combination of t h e

c o e f f i c i e n t s i n region I.

- 14-

By in tegra t ing t h e equations

HH- 5

-kd/ 2 (j s i n ~ 1 ~ 2 ) +d/ 2 j s i n T,Z) - ( I' c0s n;zj dz - i Hz J

t h e t h i r d s e t of equations i s obtained i n t h e general f o s ~

4. Matching of H, a t r a . By i n t e g r a t i n g t h e equations

+dl2 { cos Tmz) +dl2 I 1 coa C i "e j s i n T&z dz = He s i n .v&z dz

t h e four th set of equations is obtained i n the genera l forn

V I . An Approximation f o r t h e Frequency Condition.

The matching of t h e 4 f i e l d s gave 4 s e t s of l i n e a r homogenous equations f o r

t h e 4 s e t s of coef f i c ien t s . The s t a r t i n g f i e l d equations were w r i t t e n i n a way

t o make a l l c o e f f i c i e n t s and equations r e a l . It i s poss ib le t o e l iminate t h e

c o e f f i c i e n t s cm, ca, dm, d& of region 11. Then only two s e t s of equations f o r

t h e c o e f f i c i e n t s ai, bi of region I w i l l remain. They may b e w r i t t e n a s

Each element i s t h e sum'of an i n f i n i t e s e r i e s

Q)

'n i = C U e t c .

m=O nim

The expressions f o r the elements 'ni* "ni ' 'ni ' 'ni

a r e the re fo re r a t h e r

long and they a r e here omitted. A vanishing secu la r determinant of t h i s

system of equations provides t h e necessary re la t ionsh ip between guide dimensions

( a , b, w, d), kfave number k and propagation constant 80 . Approximate so lu t ions fo r t h e frequency condit ion a r e found by l imi t ing

. t h e number of terms i n t h e Fourier s e r i e s fo r t h e f i e l d s . A f i r s t order

approximation is obtained by taking n = i = 0 and . m = 0 only. This is

equivalent t o take i n region I t h e fundamental HE and HM component and i n

region I1 t h e fundamental TM-mode. The matrix of t h e de f lec t ing mode ( v = 1)

is reduced t o 4 elements:,

Jl (Y a) d J2(Ya) Zl(ka) 2 . U $a -.-- - + -, (ka) (pa) --.- . .

00 Y a Coo

J (ya) V = ka + (ka) C 2

00 Y a w ( 00

2 J 2 ( Y 4 J1(ya) X = (ka) -

00 (va) Ya

- 16- HH- 5

It is possible to demonstrate the physical significance of the

equations obtained in the simple case of the "smooth approximation", i.e.,

if the waveguide has a very large number of infinitely thin irises per vave-

length. In this case

d - w 3 1 and Coo 41.

The equation

may then be obtained directly by exerting the boundary condition E e = O at

r = a on the two fields in region I.

The equation

Uoo a. f Voo bo = 0

is obtained by impedance matching at r = a according to

The remaining HZ - component in region I has to be matched by currents in the irises.

The approximate frequency condition is given by the relation

After some manipulations, this expression takes the form

- 17- HH- 5

A t cu to f f , i,.e., f o r B = 0 , the expression takes t h e form of a product,

The two f a c t o r s correspond t o t h e TE and TM1 mode: 1

In t h e l imi t ing case f o r v = c, i.e., f o r y = 0, t h e frequency condition ph

takes t h e form:

VII. Normalization

To achieve simple nota t ion, t h e f i e l d s were developed i n terms of maa-1-

ized functions. The question of normalization is without importance f o r t h e

1 s t ' order approximation obtained. The validsty of t h i s approximate so lu t ion

was checked by comparison of computed and measured values. The i n f i n i t e

determinant, however, has t o s a t i s f y severa l condit ions [17] t o g ive a

convergent so lu t ion , and a proper normalization i s necessary.

VflX. The Power Flow

The time-average t o t a l power flow PZ through t h e waveguide i s given by

S i s evaluated a t z = w/2. z

Using t h e approximation of (VI) f o r t h e d e f l e c t i n g mode propagating with

l i g h t v e l o c i t y , t h e c o e f f i c i e n t s a. and bo are r e l a t e d by

The equivalent e l e c t r i c f i e l d s t rength Eo i s given by

Af te r some manipulations, t he power.flow through t he waveguide i s found

t o be approximately

o r a l t e rna t i ve ly

The power flow assumes nagative values i f

ka s

The maximum def lect ing f i e l d strength E i s therefore determined i f t he 0

klystron power P i s known. The power transported by t he higher harmonics z

w i l l however reduce t h e ac tua l def lect ing force.

HH: ab

Date: November 2, 1962

Distr ibution: AD B 1 , B2

-19- HH- 5

Table 1 Transverse and H7brid Solutions

2 v t l 1

k r

2Vf1("+l):

Table 2 : Hybrid Solutions for v = c eh

HH- 5

APPENDIX

A l l abbreviations used i n the main body of the report are here defined.

1.. Definit ion o f propagation constants.

In region I:

In region 11:

2 . Definit ion of integrals .

1 i. - '? e kj Snz j d

s i n . TI m zdz - - * 'nm

HH- 5

3. Definition of cylindric~l~functions

The same definitions are valid for the primed values I'A .

-23-

REFERENCES

HH- 5

1. ~ . K . H . Panofsky, In te rna l Report S~/7855/nc; CERN (1959).

2 . H.G. Hereward, In te rna l Report PS In t . TH58b8; CERN (Dec. 1958).

3. :J.K. Panofsky, W.A. denzel, Rev. Sci. Ins t r . Vol 27, No 11, 967 (NOV. 1956).

4.;' 4 . K . Panofsky, In te rna l Memorandum HEPL-82, Stanford University (May 1956).

5. P..R. Ph i l l ips , Rev. Sci. I n s t r , Vol 32, No 1, 13 (Jan 1961).

6. P. ~ a ~ o s t o l i e , In te rna l Report PS/Int. AR/60-2; CERN (March 1960).

7. J.P. Blewett, In te rna l Report JPB-9; AD; Brookhaven National Laboratory (March 1958).

8. J.P.' Blewett, Proc. In ternat ional Confer. High Energy Accelerators, 422, CERN 1959.

9. R. Gabillard, Unpublished repor t , Paculte' des Sciences de L i l l e , France (Mai 1961).

10. IC.L:, Brown e t a$r'., Proc, Xnt.ernationa1 -Conger,. H&gh Energy .Accelerators, 79, ,

Bsookhave~ (196%). -

11. H. Hahn, I n t e rna l Report HH-4, AD, Brookhaven National Laboratory @lay 1962).

12. A. Nisbet, Proc...:Roy, Soc. Lond, A240, 447 (1957).

13. G. Goubeau, Electromagnetic Maves (edited by R.E. Langer), 311; The University

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