Frequency-dependent TE-TM-mode conversion in isotropic, inhomogeneous waveguides

Frequency-dependent TE-TM-mode conversion in isotropic,

inhomogeneous waveguides

Citation for published version (APA):

Tirtoprodjo, S. (1962). Frequency-dependent TE-TM-mode conversion in isotropic, inhomogeneous waveguides. (Technische Hogeschool Eindhoven : Afdeling der Elektrotechniek : rapport; Vol. ET 4). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1962

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T E C H N I S C H E HOG ESC H 0 0 L E I N DHO V E N

FREQUENCY DEPENDENT TE - TM MODE CONVERSION IN ISOTROPIC , INHOMOGENEOUS WAVEGUIDES

b Y

1

-I. INTRODUCTION.

Based on prev~ous work by Marcuvitz [1] and De Hoop

L2J,

the present author has reported earlier on the theory of the isotropic, inhomogeneous-ly filled uniform waveguide

(3J •

V.V. Nikolskii has described a similar method, called by him "the method of eigenfunctions" t . in a series of articles

[4] •

,We shall adopt this name in this and forthcoming reports, as the metho.d is I

indeed based on a description of the electromagnetic f~eld in terms of eigenmodes of the homogeneous perfect guide, the inhomogeneous

f.

=f(x,y) and JU=;Uex,y) causing the coupling between these modes, subject to

Maxwell's equat~ons e x and y indicates the transverse coordinates) • One of the main advantages of the method of eigenfunctionsconcel'lns the boundary conditions (on the perfectly conducting wall) being automatically. satisfied by the field, as each' term in the linear expansion satisfies 'the boundary condition.

An extension of the method to. the case of anisotroJ?ic media ha.s been· accomplished by Jeuken

t5].

We shall now proceed with a short recapitulation of the relevant results and formulas of the earlier report

f3J.

It was found there, that the propagation-characteristics of the field are almost entirely governed by the so-called

A -

matrix,

'AI

=

l

z

J

IY/ ,

where

Z . .

jWP .. 1

J

= + _{jW Qji} J1 J1

(i)

.1

s

Y.'. ₌ j 1.I.);i •• _{+ . jU) ji}

J1 J1

the indices j and i designat~ng two different modes out of an in-finite number of discreet modes; while P,

Q,

Rand S are

area-integrals over the cross-section of the. guide •

, p, \ ! ' ~ , , , \ ' , 2

-In the empty guide (€. = f.(J t.

it

= ~D) we find t that the cross-coupling

terms on account of orthogonality, vanish P ..

=

Q .. = R .. = 5 .. = 0 •

J1 J1 J1 J1

In this homogeneous case the so-called selfcoupling terms will have the following values

.

.

P .. P ..

₌

j£fJ

R jj R .. €.O = =

=

JJ 11 11 -2 k2. k2. k2. k . Q j j ....sJ. Qii C1 _{5 .. ....sJ.} 5 .. C1 = = ; =

=

ED

'

,

eo

JJ

_j40

11

_/i'o

The k2 _c are of course the wellknown

eigenv~lues,

dependent on cross-section and index-number; furthermore Q = 0 when a TE ~ mode is involved, and S = 0 for a' TM,- mode.

, The diagonal elements of the A - matrix are then the only remaining elements, and they assume the easily recognizable form

z 2 ' t

k2 ' t.

2-- to f.()~+ _kcl

'.

,

-Wit

, D # + _c2

.

,

- to,

tiljtl

+ kc3 t

.

etc. etc •

In the case of the inhomogeneously filied guide with ~

=

f (x,y) and ;U=;U(x,y), tbe permittivity and permeability thus in, general being functions of the transverse coordinates, coupling between the modes wi.ll appear as the cross-coupling terms are now non-zero •

The selfcoupling terms, and as a consequence the diagonal elements of the A - matrix t 'will also be modified by the inhomogeneity.

The complete expressions for the integrals are as follows ([3],appendix A):

P .. =

J~jl

-

e .• e.

-

dx dy ,J1 J 1 Q ji =

fl

I ( Vtee j ) ( V\.ei ) dx dy .

())

])

_E

R .•

fiE

J1 = _1J fi .• _J ni dx dy 5 _.

f!

I (

_V

_teiij) ( Vteiii ) dx dy = J1 ]) _/IL

...

,> .

3

-is the transverse nabla-operator

.

,

i., i,

, . , ,. ;

J l. J l. are the orthonormal electric mode-vectol's, respectiflil vely magnetic modevectJ)rs for

\ . the j-th mode resp. i-th mode .t as

defined by Marcuvitz [ I J .

The propagation-constants ,.., of the modified modes (sometimes

~'alled

quasi-modes when weak coupling is assumed) can then be found by adding.

_ r2

_{to each diagonal element of the A - matrix (in the loss-fr~e} case ,.,2 _{J •}

=

J.2 A 2 _,~ = _ _I~ (.;2, ). , _{an So Vl.ng} d l ' th _e d _{eterml.nantal} . equation •

To each root ('r'2) cODresponds an eigenvector, the form of the

latter indicating the composition and apportioning as regards the linear combination of eigenmodes, which is then none other than the correspon-ding normal mode of the inhomogeneous guide •

These normal modes are complet~ too, so a "pure" eigenmode field-configuration of the empty guiae (for example the dominant mode) t

which we excite at the sending end of the inhomogeneous guide, may be

o~composed into the·new-found normal modes, each wit~ appropriate "amplitude" •

As these latter modes are completely decoupled and for each of them the propagation constant is known, the problem of the isotropic, inhomoge-neously filled uniform waveguide may be considered solved in principle *,

- 4

~

DEGENERATE MODES

When two different modes of the homogeneous guide have the same eigen-"

value k

c and therefore have the s.ame phase-constant

f

we call them degenerate modes • .

for any fre-quency ,

In .the A - matrix this means that the .diagonal elements associated with these modes ,are identical.

For example, the TE -mode and the

nm TM nm - mode of the rectangular'

waveguide are degeneratesQ are the TEOI - mode and the TMll~

mode in the circular waveguide •

If the la.tter guide is bent, the diagonal elements of the corresponding A - matrix remain identical, but the bend causes coupling between the TE_Ol- ~nd the TMll-mode. That is, the off-diagonal elements in the A matrix will now have a value different from zero •

It is well-known from coupled~mode theory and it can be easily proven, that degeneracy when accompanied by coupling, has a catastrophic effect on the propagation of a "pure" mode •

Even very weak coupling (i.e. very small values of the' a,ff-diagonal . elements in the. A - matri~) cannot prevent the total' conversion and

- " I /

periodic re-conversion of one mode to the other, provided of course, 'that the guide-length is adequately long •

We ma)t conclude then, that in the case of the diagonal elements of the A - matrix the off-diagonal elements only affect the

. '1

degeneracy, thenequal-ness of are all-important, whereas

so-called criti~al guide-length.

There are two ways to avoid conversion J or rende r t'-t. 'j'1tsj11.--t.ft.' dd},?1.

1-The first is to remove the degeneracy; in the sense of our formalism this means making the diagonal elements different from each other •

We may achieve this by putting a thin dielectric layer on the inside of' I

the guide-wall, as the TMll-mode "reacts" to this device in a different manner ,:til: as compared with the TEal -mode

(ilt'

the..

betk.·f, rbVlJ1."el w.3

vet lI/:.teJ.:.

I .

,

. 5 .

-The second method consists of filling up the circular guide inhomogeneo

_l1

s-ly in such a way, that th.e off-diagonal elements, i.e. thecroaa-coupling .. integrals, become

identic~l

zero ( 6

J .

Degeneracy has theoretically no effect now (in the. two-mV~Q appr~Ximation)

but as we cannot in general avoid indirect coupling via other modes , the first method

[7]

is to be pr.eferred •

What we intend to do now, is to investigate the "duapt problem • We seek an answer to the following paiD question

I

if we have two

.!!£E -

degene~ate modes of the homogeneous waveguide, thus

'"

having different values of their appropriate diagonal elements in the , A - matrix (for any frequency) .; can we or can' t we find an inhomoge- .

. ' -,

-

.

neous configurati.on in such a way, that.the diagonal elements associated with these (now coupled) modes become idp.ntical ?

'It will be shown, that th~ answer is in general in the affirmative, subject to the .restriction that degeneracy will only occur at

one specific frequency

The underlying reason for this investigation is our expectation,thatit may help explain some experimentally found "anomalies", e.g. the peculiar behaviour of the attenuation curve of the coaxial-cable with spiralized dielectric as centering device, at some frequencies

[8

J

~

On the other hand, siMilar "anomalies" are to be expected for special inhomogeneous fillings and at one specific frequenc, or frequency...;range (backward waves ?) , and these may now be theoretically predicte'd. "

6

III.

THE INHOMOGENEOUS RECTANGULAR WAVEGUIDE

-...

-Cons ~der the ,above waveguide ( b

=

-fa )

_with

}'=/,o

and

E=

E(x,y) •

Assuming weak coupling and restricting ourselves to a two-mode approxi-ma tioD'; ass,1..gnindex 1 to the dominant TEIO-mode and index 2 t,?

the lowest possible TM-mode 'I'M I 1 (2 x 2) A - matrix become then

the diagonal elements of the

2 7['t

}

-W'j<c

R + .a~

(3)

() 11 fl R22 Q22

- WfUo

R22 +

"

In the ,empty guide _RII

₌

_R22 =

Eo

Q 22 I k2 1 57["z

=

=

Eo

c

Eo

2 a _•

T-tL

th!

(. 'k. It 0

'Ik.Cl/e

4t-cZ Of/$'

j

U (de: _7[2

-iii. th R

22Q22 apparently much larger than d.,tl if R22

<

RII

no real frequency exists, for which th. two expressions ( 3 ) have the same negative value

.

If on the other hand, _R22

::>

_RII

,

then however small the difference may be, d egenracy e is assured at some specific frequency (which in the following we will call i!xx fd ) .

The larger the difference between R22 and RII the lower will and vice versa.

i.

7

-The question now arises; if we have at our disposal a thin or small homoge· neous.dielectric slab or rod, then where do we have to put this slab or rod (alongside the axis of the guide), to ascertain, i f possible without trial and error, tha t the condi tion R22

>

Rll will be met ?

We recognize that this looks like some sort of synthesis-problem, but it so happens that a brief consideration of the R- integral will supply us the necessary clue •

r( .

=

J) £. (x,y) ]) =

f[z

(x,y) J)

hl,h

_l

h

₂

·h

₂ dx dy .dx dy •

}

' , , •

Now, i f ~ = Eo (constant), then . normalization - rule:

ff

hl·li_l

Rll

=

dx dy

=

R₂₂

=

to on aceount of the_

=

1 • I> The TN, - mode

11 has a, zero transverse field at the centre of the

guide as Xxx for (x=ya,y=yb) •

Thus, placing the dielectric rod in the middle of the guide will not (or almost not) contribute an increase in' the R22 - value ---.:;;.. R22 ~ ~ ... The TE

10 - mode however, has a rather strong field at the c~ntre, and as a consequence Rll> £0 for the above placing of the rod •

Of course, for our purpose (R

22

»

Rli) this 'is the wrong .ay, but the principle behind it is obvious

if we want some R - integral to exceed another, then put the dielectric in that region of the cross-section where the transverse field of the mode associated with the former R - integral is stronger than that of-the latter.

8

-that the transverse field is, 'so to speak, pushed to the sides

t

For the TMll ~ mode more to the upper and lower region than to

tee vertical sides (fa" the TEll -mode just the contrary )

As regards our aim to achieve degeneracy b,etween the TE

IO - and TMll -mode. following the forementioned considerations, we then decide on the

inhomogeneous configuration, as' sketched below

E

=

~I , f

t

o<x<a.

or . 0< y< $b

'/

l'

for

This slab-form has the added advantage that the .TEOl-mode ,will, not be . coupled, as the TEOn-modes form an autonomous group for'the above

'confi-guration ( [3J, p.24 for an analogous example) .

Some simple integral-computations will get us ~he folloWing results

.

.,

.a

p

£0)

r

fll

.1.£

tey)

hI-hI dx dy

=

z:..c+ (£,

-R22 ,;; IiJ,E(y)

h

₂·h₂ dx dy

₌

e

_o + ( E, - ED)

d

+

~(€/-

eo) sin 2714c

. a Q

(here Ibc: _hI

₌

-p ~ 5~ 7(.X •

a

hID =

a

a

x

h2

=

hll -q

=

~

W;

S~ 1Q. CUd

!0/.,,a

~

V!

i!<r.f 7C.X S ~

7lY -

b

.d

_y

. S' 4. b x d $"". d

.i

9

-Comparing 'R

ll wi th R22 assures us of the condi tion . R22

>

Rll always being fulfilled, as soon' as

E:.,.'"

1 and K>o _•

Furthermore, it may be verified that the cross-coupling off-diagonal element Al2 of the A - matrix is non-zero (bp.cause Rl2

F

0 ) , so at some definite distance from the guide-entrance near-total conversion from the TE

10- inti) the· TMll-mode is to be expected, at frequency fd • We remark that the selfcoupling H-integral of the TEll-mode i6 always smaller than. Hll ' as it can be calculated to be

Eo +-

(~l-

(0)

0

~ 1~7[

(£,-

fo)

~

:Ul"tJ.

The TEll-mode therefore will never reach degeneracy with the TElO-mode,.

To clarify some of the results. it seems best to proceed with a numerical example. Consider the configuration of the preceding page

in order to limit the coupling somewhat (so as to prevent too much conver-_, sion to higher modes than the TMll-mode), take the P7rmittivi,ty of the slab to be E, = 2 Eo

---From ~he expression for the R22 - integral it is clear~y seen that

F-

value·

and

will produce the largest possible difference betweeL

impl~ing a frequency of degeneracy fd as low as possible.

For the above circumstances we will then aquire the following results e RlI' =

t

£, +

i

Eo

=

1 t

Eo

R22 (t f,

i

Eo)

. l .,

=(

It

3 = + + /0 7[ (f, - Eo) + 107[ ) Eo .a. j,

=1£

f~Y)

V

-q 2 Q_{22 ( t·ell)} dx dy = 57[:2.

ITL~

[3

I . ' ( ' I

J]

I

(f

+

;~)

=

_a

L

+-+

--:-

= _{. ~o}

_J!2

lifo t{f, :l.7r:. £.0 - £/

Equating the two diagonal elements of the A - matrix, i.e.

•

\

;-- 10

,

we find

w2..t:

c),-o ,:

7[;

=

7['

i

r

(I:

7- ,:

lC)

(f

+

:1C) -

1 }

2..

if

7fl

As

t.U

Eo

j'-o=

A;

wi th ).0 the :Our: free-space' wavelength"

we obtain for this configuration, the following wavelength of degeneracy -:

A

= .,0,265 a, %

t

a

o •

cis •

(with a the largest transverse guide-dimension Giorgi system of./units)

A table of the values of the diagonal elements All and A22 as a function of ~ ₀ is given below

.

f

it"

= 0, >'

.a

AD::

0,,/ d Ao.=0 .. 3 J. ;{ _e., .:: _0,25"4 /t-( 1 - O,2..a

Rn

7[.(.

3c:,t 7[t. 57;~S' 7['2-

7j

7C-t 1.2'f 7L<

-:>

It]

_{a,,z d-< .a~ d..t.}

_.a.

_{R- •}

A.22~

;,

-

7[2 _1.1,2. 7C2 _5"~

_7"

~

-

go

7C~ l.z,f 7l'~

d Z

.a...a

d.t d..t..

,d.t.

\

Having found f d we next consider the ques·tion of the _ conversion reconversion effects and of the guide-lengths involved, for the above example ...

The value of the off-diagonal element of the A - matrix may be shown to be almost entirely decided by the cross-coupling integral

'1

a

.6

'

R12 =

r

E (y)

o

J

o

We remark once more, that inserting the dielectric in a special way (synthesis!) may result in R12 becoming zero, whereas the self-coupling Rll ,or R .. in general; can never be smaller than

_Eo

JJ

-1.

18t

7["1-we find _W

_/,,0

_R12 =

a.z..

• With

A

~

t

a o

In the case of degeneracy it is well-known from coupled-mode theory and

,

i t can be easily proven by the method of eigenfunctions, that in the two-mode approximation the new normal-modes may be thought of as combina-tions of equal parts of the two modes involved ;

(symbolically) x.!x~xlix

"t

TE IO +

t

TMll II and in our case II

t

TE -

t

TM II 10 ' 11' , , •

11

-The difference in sign in the symbolic expressions, indicates the fact, that at the entrance of the guide ( z

=

0 ) , the TMll - parts are in anti - phase, as for z

=

0 no TMll-mode field is asyet.presept, just the incoming TEIO-mode

These two approximate normal-modes will propagate with different phase-constants

(5/

and

132

which can be calculated as follows

!/~

.2

_-rc

z

7C-<-7L

I"V

78i

ZL

18i

/'1

/"V

"-' _{dot +}

a-t.

₌

97

_,dZ ~ A...- .

10 .a

}/

rV

78t

7l:

z

18i

7[2. .

60

-,z:< - - 7

134

~

₈

7L.

I"\..- _a.l

=

_{2..2 "'-} ,

.

d-Z

.a.

Near-total conversion from TE

IO- into the TMll-mode will then take ,p.iaee / for the first time at a distance z from the entrance, corre~pon~ing to

=

7C

~

( 10

7C _

.:::t

Thus, at the points z

=

f a , z

=

I i a , z

=

2i

a , e t c . ; we may ex-pect regions of predominantly TMll mode-configuration' in phase !")

For instance if we take a

=

8

cm. , quency must have a free-space wavelength

the operating

fre-- " .2 cm.

The periodically spaced regions where the incoming TEIO-mode will be almost totally converted into the·TMll""mode may then be found in the vicini ty of the points z

=

4 cm., Z = 12 em.', Z = 20 cm, etc. etc.

As the TMll-mode i s a higher mode compared with the dominant TEIO-mode, from the viewpoint of eig~nmodes of the empty guide, there apparently exist alternating regions of higher and lower phase-constantsj (at f

d) • This means, that as we proceed from an operating frequency slightly lower

than f

d, e.g."Cfd-,Af), to fd, the fieldconfiguration in some fixed re-: gions will change from predominantly TEIO-mode to predominantly TMl1-mode. For these regions of the inhomogeneous guide then, a decrease in phase-constant has been effected in conjunction with an increase in freguency , as a consequence, within a narrow frequency - range , backward -waves seem indeed possible •

,-•

i ,

12

-We realize now, that for the occurrence of periodically spaced "backward wave" - regions, we do not have to restrict ourselves to TE-TM coupling Any lower mO,de - higher ,mode conversion, with lower mode as input wave and the' higher mode more to benefit from the special way of inserting th dielectric, will eventually reach degeneracy at some frequency fd

However, TE - TM coupling is thought to be more feasible for experimental verification •

One final remark on the re~tangu1ar guide: if we want to avoid TE 10 :";, TM11 'coupling for the above example, this can indeed be don,e by simply slicing the slab ,horizontally in two equal parts and move one part 'to the upper wall. For this symmetrical configura'tion, a1 though the condition

R₂₂

>

R11 still holds, now the cross-coupling R12 - integral will be

identically zero

IV.

TD

INHOMOGENEOUS

CIRCTTLAR

WAVEGUIDE _•

As the underlying principles of the theory are assumed to be reasonably understood by now, we give the results for certain inhomogeneous confi~

gurations'of the,round waveguide without much comment.

,Only the frequencies

ot

degeneracy will be calculated here. Further de-I

tails about' the periodic distances of alternating conversion-regions will be published in subsequent reports •

The first configuration to be examined is the circular symmetric case of the round dielectric rod placed a1~ng the axis of the guide

jl=jlo

, p..::.:

1. :'

, £ ::: f

(r) .

for 0 < r <.

td,

" pd <::.

r

<t..

a,

i

•

I '

13

-We want the dominant TEll - mode to couple with the lowest possible TM-mode and the two quasi-modes to reach degeneracy at a specific frequ~ncy • For this configuration the TMOn-modes (and the'TEOn-modes for that matter) form an autonomous group, so the TMOI-mode will not be excited, the TM

I1 -mode then being the lowest TM -mode coupled to the dominant TEll--mode. Assignate index 1 to the latter, and index 2 to the former (TM

Il) mode We can be reasonably sure of the condition R

22

>

Rll being satisfied, , as we may easily verify that at the centre the TMll-field is slightly stronger than the TEll-field

n

We obtain the following results after some elemtary computations

-~L

Unlike the former example of the inhomogeneous recta~~ular guide, here the condition R22

>

Rll is not fulfilled for' all p> 0 (. pa being the radius of the dielectric rod, a of the guide itself ) .

Certainty can be had however, by sketching K and L as functions of 'p • For those values of p where

_L>

_{K holds, so will R22} ~ RII Even here we may predict the outcome •

The TM

ll - fiJlxli mode has zero transverse field at some points cor- ,

- responding to p ~ 0,6 therefore the advantage of the stronger field at the centre will surely die'out for a value ofp below p

=

0,6.

We may check first

,

_{that for the above RIl}

-

and R .- expressions 22 when p

₌

0 --';10 K

=

L = 0 ~ RIl

=

R22

=

Eo

when p = 1 ----?l- K

=

L

=

1 --"> _R1I

=

_R22

=

_f/

as i t should be _• (1?-12:;/:.·0 J

for

0 < :

P

<

i)

~

..

14 j

X ,L

/ ' _.MII _

_/

'.~

t

}</

/1

K L '.1 p = 0,1 0.02 0 .. 03 I;

I

p

=

0.2 0.08 0.15 / I

L

p = 0.3 0.17 0.21 'p = 0.4 0.29 0.29 p

₌

0.5 0.42 0.35 p= 0.6

0.55

0.41 0.7 0.71 O.Sl tI, P .= P

=

0.8 0.80. 0.63. p = 0.9 0.90 0.80

".

/ tI,tl. D,J tI.'1 V ·1>,4 '1,7 D.

r

D.~ J ~f>

From the above diagram we conclude, that dielectric rods with radii larger than 0,4 a will not satisfy the condition _R22

_>

_Rll ; < i.e. no

dege-neracy between TE

l l- and TMll -mode is to be expected,. at whatever

frequency •

Clarricoats arrives at the opposite conclusion in his paper ( r9J'iig.3~:

b~t nevertheless his diagram and ours bear

a

striking resemblance as to

the critical point p

=

0,4 •

As the difference between K and L (for p<.0,4) is greatest for p

=

0,2 we take as numerical example a dielectric rod of radius 0,2 a

mittivity !-L=_?~5!.

'

as before.

and a pel

A table of the values of the diagonal elements All. and A

22 as a

func-tion of the free-space wavelength

'1\0

is given below

A,,::::

o,6a.

Ao::

0, :;-~.

.::to::'

O. V.J. . !

A

4 ·"'oa~ > •

1I

1

/d2

;

'to

/a.t. .

2b7/d-<

'17 b

/a.z .

- -

-

- -

---.~~-liZ / "l2 .

/b8/

_d.l ; Z}2/d.1.

~°7"4z.

.-The specific frequency fd can be computed to be

( for this case: Rll = 1,08

Eo

and _~22

,

.

15

-As regards the dual configuration sketched below

/~o·.

,,<.

t .

c.::;

f(,.) .

for

0 ...:: ~ 4

fa. .

't

pa

<

r.,c:.

a,

a circular waveguide with concentric dielectric layer adjacent to the'

I

we. 1 1 , we remark, that. the TMll-field close to the wall is indeed

slightly stronger than the. TEll-field'; 1. e .. favour'able for conversioll For the selfcoupling R - integrals in this' case, we obtain

a .rrc:

Rll

'=1'1

_{o 0} E(r) _hll·hll -p -p r dr d<f

=

E -'

I, ( £,- Eo) K

, R22

=

fj,:urc(r}

_hll-hll -q -q r dr df

=

£-

I

( £0/ -

.

E )

0 L

(7 0

" . .

K and L are the same as that of the preceding example, so we may use the diagram of page 14

The conditionR

22

>

Rll now implies the above expressions for Rlland R

22 •

Thi~ dual configuration then, leads to an analogous restriction as in

the former case " namely that now it is the thickne.ss of ' the layer , which must not exceed the value 0.6 a

,

as for the interval

p <:::. 0.4 the value for" K will be smaller than that for· , L We may check for the extreme case (homogeneous)

when p

₌

0 ---':>' K

=

L

₌

0 ~ _Rll

=

_R22

-

, E J when p

₌

1 ~ K

₌

L

₌

1 ~ Rll

=

_R22.

=

_to

as it should be ( 1<.12.. =J=

far

,~ 0 _.J O..::.p <: I)

..

16

-in the -interval· 0.4 <: p

<:

1 occurs for. p = 0.7

Thus, to assure a frequency of TEll-TM

ll degeneracy as low as possib~e (for a given permittivity), the dielectric layer adjacent to the wall

_,

must have a thickness cocresponding to p

=

0.7, that is a thicknes

of 0.3 a .'

For the numerical example we choose then:

_E

_I

=

2

to'

and p = 0.7

We find for these circumstances _Rll = 1.29

E.

₁ d2

ll

0

Q₂₂ 0.98

Eo

2

R22 = L49co ,a·

A table of the values of the diagonal elements All and A22 as a function of the free-space wavelength is given below

Ao:::

o.a

a

Ao;:::

O}2..

Ao'"

O,/;d

R

I r -->

lb .

.t/

a::

g1'~

_{d.t .}

.

ILjO.

Ja.z

--_.

2 ~

11. hlat

gl'1fa~

Iff·.?!

_d.t

The specific frequency fd

can

be computed to be

;1.0==

O.S-

a

z(Jo'h~

Jl/6.~

_d.t.

.

3.l{)8 cIs o,66a From the above examples it will be clear by now, that a dielectric insert may cause degeneracy between a lower and a higher quasi-mode . at come specific frequency· f d ..

Now, the preferred solution in the bent circular waveguide (fortele- e communication purposes) is to remove the TEOI - TMll degeneracy by dielectrically layering the wall •

The TEO 1 - field adjacent to the wall is zero while' that for the TMll-mode is non-zero, therefore, removal o~ the degeneracy is .assured from the beginning (cut-off) and will be permanant, as, the

W-f.

cnarac-teristics for the two modes are divergent !

..

•

17

-wavelengths as small as a few rom. are proposed, so even a very thin layer is sufficient •

In the light of the above theory, bearing in-mind that the phase -

_.

_. constants of all modes '(except the TEOn-group) benefit more or less, not only wil'l such an extreme thinness be sufficient, it is absolutely necessary as well to prevent the. next higher coupled mode (TE

12) reaching degeneracy with the TEOl-mode at .some fd within the interval of operating frequencies, thereby spoiling the favourable attenuation - curve of the latter •

As a final example, let's have a look at the following sector-slab adjacent to the wall of the circular guide, a configuration which we believe has far-reaching implications in view of its similarities as regards the spiral coax.

-for

(}<:.

f

L. c:( . ,/ 0{

<.

ft

<...

27t:.

For. this inhomogeneous filling no circular symmetry exists, so the TMOn-modes no longer form an autonomous group, and the lowest possible coupled TM - mode wkll then be the TMOl-mode

The electric field of the iatter is radially directed, hence, as re-gards the associated ph~se-constantt will certainly benefit from the dielectric insert •

To assure the occurrence of degeneracy~effects, we choose as

polariza-\

tionplane of the inc')ming dominant TEll-mode that one which CO.Dres-ponds to horizontal electric fieldlines. The reason is obvious

..

,fii

- 18 ..

this TEll-mode gets almost no benefit from the insert, whereas the other independent (orthogonal) TEll-mode does profit •

Assignate index 1 to the dominant mode, index 2 to the TM01-mode~ t

we find after some computation, that our expectation concerning indeed comes true

The calculations involved, are a.o. as follows

-q hOl

=

R Z2

=

with In this expression 1// =

01

..

.

'

19,

-Degeneracy will be possible, as soon as

cL>

0 and

We

may check, as before R12

#

0 can be verified. :I when

d

=

0 ~ Rll when

d

=

:fJ 27[ ~ Rll = =

For the largest difference between Rll and R22 we must clearly have the value of

c:l

forresponding to 2

d..-

=

lC/2

; d=

_{_}

'7l:/4 rad.

....

_---In the numerical example we choose, conform the preceding cases: ~,-::~.

We obtain then Furthermore dOl::' 2.11 ' Hence, R 22Q22

(;;.)

.

A table of the values of the diagonal elements All and A22 function of the free-space wavelength

do

is given below

as a

I

ACl

-=

a

;10

%a

/l.~

=

%4

A,,=

y;

2. d. d,/~ 3').

9/

a

:t

5"1-%,,-

flLZ

IGj,%

II .;- ....!!.-<;?

a..;t

d.-Z

:a~

6

~;

_{J13~ ~}

"-.;.;6

3.z,%~ j"7(,~ tl - 1.0 _dA. --+ , 4

:,a,Z

. 'd..'l " The simplest 'solution to avoid or render insignificant this TEll'-TM

Ol total conversion at the frequency of degeneracy fd is, of course, to ,use the other (vertical) polarisation of the TEll - mode.

..

20

-.

However, should we slowly spiralize the die+ectric sector-slab within the guide, then this solution will not do, as the incoming TEll-mode of whatever polarisation will periodically encounters lengths of guide

favourable for' conversion into the TM

ml- mode at some fd

Only one method avails: it so happens that the TMOl-mode has zero transverse field at the orilin, hence what we may do is to provide the slab with some extra dielectric located at the sharp edge. Le. at the centre of the guide. The TEll-mode will strongly benefit, the TM

01 -mode almost not at all.

occurs.! One'possible reason for avoiding TEII-TM

Ol degeneracy XK when lengthy guides are involved, as the attenuation curve of the TM01-mode (due t~

wall currents) is considerably worse compared with the TEil ([lOJ,p.~rn

Now· this is precisely the link with the spiralized dielectric of the coaxial cable mentioned earlier. As communicated privately to the author by ir.C.Kooy the· experimentally found anomaly consists. of a rather sudden and unexpected jump of the attenuation at some frequ'ency.

This spiral-cable can be conceived as an inhomogeneous waveguid_, with eigenfunctions and eigenmodes associated· to the:empty coaxial cable. In subsequent· reports then, we will investigate the possibility of the above anomaly arising out of frequency-dependent mode-conversion

(degeneracy) , the TEll - mode excited by the dominant TEM~mode

.,t/U'k-to tally converting its energy in .,t/U'k-to the TMOl -mode at some frequency f d ,. thereby causing a jump ~n the attenuation (stronger wall~currents) Anticipating somewhat, we may prop9se the following solution

to suppress the anomaly •

/ I -,;:;'"'" ::-/ /

21

-This eventual solution is based on the same kina of reasoning as used in the preceding example •

The transverse electric field configuration of the TM01-mode of the 'coaxial cable, which is a radial one just like in the circular guide~

has a concentrical zero-region about midway between outer and inner conductor •

The precise location can b~ calculated for any given dimension of the coax, making use of the formulas (combined B-lssel-Neuman,n functions),

as' published in the Waveguide Handbook (Marcuvi ti) t page 76 •

Hence, attaching' concentric dielectric fins (see figure) just at the regions of the'spiral, corresponding to the zero transverse field of the TM01-mode, will not enhance the phase-constant of the latter,

. ,,_ : ' - . r . , .

whereas theTEll~mode benefit from this dielectric additions, on ac-, cac-,ount cf the non-zero TEll transverse fiac-,eld there • (The W -

j3

,diagr~

of th~ TEll gains a little steepness, that for the TMol not at all) (of the asymptote)

As a consequence, the quasi TEll ~ quasi TMOlmode-degeneracy (for a specific frequency) caused by the spiral , will now either be suppressed altogether or "delayed" to far higher frequency:

,At the end of this preliminarY,report, the author wishes to.express his gratitude to drs.M.Jeuken, ir.C.Kooy and ir.W. vanVeenendaal "

,

.

~ .. i' L i t t e r a t u r 1. 14arcuvi tz ,

N.

2. De Hoop, A.T. 3. Tirtoprodjo, S. 4. Nikolskii, V.V. 5. Jeuken ,. M.E.J. 6. Morgan, S.P .. 7. Unger, H.G. · 22 -e "Waveguide Handbook It unpublished notes.

"Golfgeleiders, gevuld'met een isotroop inhomogeen medium" • (intern rapport ET 2 - 1962) series of arti~les in

"Radioteknika y Elektronikaft 1960~1'uJ... _{/ .}

"Golfgeleiders, gevuldmet een. 'anisotroop medium.tt _•

(intern rapport ET 3

:. "Curved circular waveguide, containing an inhomogeneous dielectric" •

(Bell System Technical Journal , vol. 36, (1957), p. 1209 - 1251).

ftDielect~ic- coated Waveguidett _~.

(B.S.T.J .. , ·vo1.36 (1957), p. 1252-1278) ~

8. Van 't Groenewout, H.W.F.: "Co8.xiale kabels met spiraalvormig dielectricum" •

9. Clarricoats, P.J.B.

10. Stratton, J.A.

(Tijdschr.Ned.Radio Gen.,vol.24,nr.2/3.l959). ftBackward. waves in waveguides

containing dielectric" •

(Inst. El. Eng., Monograph 451 E, June '61). "Electromagnetic Theory" •