The Iris-Loaded Wave Guide as a Boundary Value Problem. II

The Iris-Loaded Wave Guide as a Boundary Value Problem. II

WOLFGANG KROLL

Department of Physics, National Taiwan University, Taipei, Taiwan

(Received 1 May, 1965)

In the previously published paper we sketched a theory for the iris-loaded waveguide. In this paper now we derive the complete set of integral equations of this theory.

In order to obtain from this theory a solution as proposed by Chu and Hansen one has to make approximations whose justification seems rather difficult.

In the case of large separation of the irises we obtain easily good approximate solutions from our theory.

THE INTEGRAL EQUATIONS

.

. AS in our previous paper”) we consider here an E-wave in a waveguide of radius

b with irises at distances I from each other. The radius of the holes in the irises we call a.

The Cartesian component 2% of the electric field satisfies

1 8

,s

--Y dr

( >

+

a2&

with k as the propagation constant in vacua. Because E, must vanish for r=b we solve this equation by

J(rvy) sinhdr2,-k2 z.

Here because of Ez( 6, z) =0, T, is to be found from Jo(r,b) =o.

Denoting by F1 (Y) + Fz(y) the value of E, in the plane of the zeroth iris, where F1 corresponds to an even and F2 to an odd mode, we find as the solution of Maxwell’s equations

- ikE, = einhl

c

sinhd( )z’ +~bt(F’+F2)h(r,t)dt sinh I/(

) (Z-2’)

IRIS-LOADED WAVE GUIDE ₁₁

ikE, = einhl

c

I~~

cosh,,‘i_‘Sz’-~‘t(F,+F”)Jo(Jo(i,t)dt coshd(

) (l-2’) 1.

Here h is the propagation constant in the waveguide. v”( ) is an abbreviation for 7/r;-k2. z’+nZ=z with n as the number of the corrugation.

The normalization integral

N,=ib t/;( I.t)dt=J’tJ;( r,t)dt

in the denominator is omitted.

The continuity of E, through the zeroth iris-plane leads to the condition. (l+e-‘*‘) (coshl/T-JZ-l)~btFIX( r,t)dt+ (l-e-jh’) (coshl/( Z+l) x

~btF,~&,t)dt+2(cosh~( I-cos(hZ))~‘t(F~+Fz)/,(r,t)dt=O._II

Expanding F,(r) is a series of Bessel functions we find by the use of this relation.

FI(r) +F2(r)= -2 sin’g

Jat(~l+~~)~~~-&$+!++

_ (l_e-‘“I)~btF2~ h(T’vt)jo(7,r) d t : r<a

cosh{( ) Z - l

(l+e-‘h’)FI(r) + (l- e-ihl JF2(r) = -4 sin2hl J-‘t(Fl+Fz) 1 c&(;;~& dt_{2 0}

-2(l-e-ih~)~btF2~ c$‘~~~~~l d t : r>a

To these equations comes in additon the condition that E,, the radial electric component must vanish on the iris-screens. This condition we use in the form

We introduce

and__ can then

J

the notation

FI(r) +F2(r) =F(r) : r<a

(l+e-jhz)FI(r)=Gl(r), (l-evihi)F2(r)=Gz(rj : r>a. write the continuity-condition for E, in the form

(cash ~)r-l)~btG,~(r,t)dt=2 ( cos(hZ) -cash-r/( )Z)~‘tFJo(‘.t)dt -(cash 1/( ) Z+l)~btG&(~vt)at.

The three equations take then the form

J’(r) = -2 sin2 $!-laIFC &(!~?~Z!!&~~~

Joo(r”l)JO(r”t)_{-~ pdt : r<k}

coshd ( ) Z-1 * (1)

G,(r)+Gz(r)=-4 sinz~~utFC~~~~~~~-ldt

-21btc,C /b(rvr)JO(r_{cash/( ) Z-1 *}”t) dt . r>a ( 2 )

2 sin2$ia tF 1 ~~~;~)-$‘f$$1/( (coshdm I+ 1) dt

+~btGz~$$-$;iZ ~d( ) (cosh,I(Z+l) dt=O : r>a. ₍₃₎

These are the fundamental equations of the theory. From them we have to find the functions F(r), G1 (r) and G,(r) and also the propagation constant h.

The equations can be written also in another form by elimination of Gz. Using the continuity condition for- IS,, we get

F(y) =2 cos2 ~~“t$‘~~J~~hzj?zj dt

+ ~bG~ _{cash/-j 2 + 1}_L(~oy)Jo(~vt) dt : ‘y<a (1’)

G,(y) + G2(r) =4 cos2 ~~~“tJ’~-~~(~#!j~i dt

+ lb tGIx -g&$p{$ dt :. y>a

_(2’)

+lbtGI&- _Y ‘- -__Jo(rvy)Joo(r t)

7, sinhl/( ) Z ti ( ) (cosh,/‘( ) Z-1) d t = O : ~>a. (3’)

From the last equation of the two sets we see that G,(r) will be small for small

hZ hZ

values of sin __ and G,(r) for small values of cos -._{2 2} VARIATION PRINCIPLE The equations

F(y) = -2 sin2 &L_{2 ~‘tF~-~_z =- ~--}Joo(lvy)Jo(rvt)

cash/( ) Z - l dt-~btG@&#&L;!& dt : y<u

and

F(r)=2 ~o~2~~-~otFC~~~~~~idt+~btGz~~~~~~~~ dt : y>a_

IRIS-LOADED WAVE GUIDE ₁₃

can be derived from the variation principles:

sL= $‘rFdr [1F(y) +2 sin’ _; i=tFG,-(r, t) dtf2 lbtGz Gk-(r, t) dt]=O, and 6M= bi’rFdr P(T) -2 co? +i ‘tFGk+(?-, t) dt-2 lbtG, Gk+(? t) dt]. Here and JO(r”r)/o(r”t) Gk+ = c cash,/’ ()I+ 1.

When h takes such values that Gl(r) resp. Gz(r) can be neglected, we can write the variation-principle in the form

s

sL=a ~‘rFdrjtFGk (r, t) dt.

H e r e Gk(r, t) stands for Gk+ YeSp_ Gk-. We obtain then 2 cos2 7 resp. -2 sin2 -$-as the extremal values of L, provided that (cos i I resp. I sin: I is small.

SOLUTION FOR LARGE SEPARATION

When 2 is large as compared with b, in the equations (i) and (1’) we need only the first few terms in the series GR- and Gk+, the higher terms going to zero exponentially.

Assuming

Tl<h<T2

we need only the first term. Confining further the consideration to small values of (sin % I .resp. I cos “2” I, the equations take the form

2sin2gj(ry) 2 o 1 F(r) = y- N(cosl/&-~;-z-l) siIatFjo(rtt)dt 2 cos2 K J (r1r) 2 _--” Fb-)= N(cOs,/&-rl Z+1) J-rJ‘tFJo(rlt)dt. The solution of these equations is

IRIS-LOADED WAVE GUIDE _{1 5} We try then to approximate

Gk-

1_{~ =--}2 cash z-1 z2 ’

Then we obtain

This is now the well-known expansion of the Green-function.

>

:

r<t

Gk(r, d)=-_ii

Jo

4vdkr)) :

r>t.

Confining the

with

consideration to small values of (sin T I we get F ( r ) = -f(h)i=tFGa(r, i!)dt : r<a

f(h) =f sin’ -“2”_.

This equation now is easily solved. We find F(r) =No(l~K~-f(h) r)

Jo(l~K2--f(h) a) No(M)J3(Fia) -No(K

_---

-,~P2_foJlh4+-f(h) a) = k(No(Kb)JI(Ka)-~(Ka)J,(~b) *

G,(r) is negligible and G1 (r) is proportional to No(K --No(Kr Therefore as far as the electric that obtained by Chu and Hansen. in remarkable points.

From this consideration it seems

field is concerned, the solution is similar to However the frequency-condition is modified that a solution as proposed by Chu and Hansen can be obtained only b replacing the denominator in Gn- by the first term of an

4.

expansion in partial fractions and it is difficult to see how this replacement can be justified mathematically.

The support of this work by the Chinese Council on Science Development is gratefully acknowledged.