ELECTRDrlAGN1'TIC SURFACE WAVES -------------------------------by A oZ .TIRKEL

A report on theoretical work carried out in the Physic s Department Monash Uni versity as part of the requirements for the Bachelor of Science with Honours Degree at MONASH UNIVERSITY in 1970.

Supervi sor: Dr.JoL.A.FRANCEY.

I N D E X.

Io INTRODUCTION 1

2 II. § 2. 01 What is a SURFACli' WAV"' ?

§

§

§

§

§

§

§

§

~

§

§

§

III. §

§

20011 History of surface waves and development in wireless telegr~phy

2.012 Note on the difference between seismic and radio surface waves

2 .02

?.OJ 2 .031

2.032 2.,033

2o04 2.041

?.042 2.05 2. 051 2.052

2.053

2.054 2.06 2.062 20063

2.07

SURFACE WAVES as SOLUTIONS of MAXWELL'S EQUATIONS MORE COlV'JVlON T:[PES of SURFACE WAVES ITuWEDANCE CONDITIONS POLARIZATION ASYMETRIC WAVES PHYSICAL INTERPRETATION of the ZENNECK WAVE Power carried by a ZENNECK WAV~ Note on other waves and SURFACE WAVFS in DIELECTRIC EXCITATION of SURFACE WAVES FIELD ORTHOGONALITY R~LATIONS EFFI CIF.NCY APERTURE LAUNCHRRS THE FLARFiD HORN - A PRACTIChL LAUNCHER EFFECT of DISCONTINUITIES Perturbation Techniques BOUNDARY VALVE PROBLEMS DIFFRACTION of SURFACE WAVES by WEDG~ and APPLICATIONS SURFACE WAVE AERIALS SLOW WAVE GUIDES COL~LED SURFACF WAVE GUIDES

2. 1 01 DIELECTRICS 2 o 102 COUPLING on a MULTI- ELEMENT SYSTEJ\f. 2.,10J APPLICATION to VISION PROCESSES

2

4

4

6

7 8

9 10

11

11

13

13

11

14

14

1 5 16

20 21

23 ~4

25 25 26

27

2 0 104 2 0 11

2 . 1"

SHIELDED WAVES 27

2.13

J. 1

J. 11

3. 12

J.13 3. 14

3.15 J.2 3.21

HARMONIC ANALYSIS and PULSE PROPAGATION 29 by SURFACE WAVES APPLICATIONS OF SURFAC.r: "NAVES TO SCATTERING PitOBLEMS 31

APPLICATIONS OF SURFAC"S WAVE TRi\NSMISSION LINES 33

SURFACE WAVES ON FREE PLASMAS 34

ASSUMPTIONS 34 SURFACE WAVES ON HOMOGENEOUS UNMAGNETIZRD PLASMA

LONG I TUDINALLY MAGNF.T IZED PLASMA WAVEGUIDE

QUASI-STAT IC METHOD B'OH SLOW WAVES ELECTROSTATIC SURFACE 'NAVES

EFFECTS OF AN OSCILLATOR MOVING CLOSE TO A PLASMA CRITICAL ANGLES

35

37

38

39 40

3.22 ANGULAR DISTRIBUTION OF SURFACR WAVR RNF:Rr.Y 42 Ll..1

§J. B INTERACTICN OF AN ELECTRON BEAM WITH A PLASMA VIA SURFACE WAVES

J.J1 AIV:PLIFICATION AND GENERATION OF sum~ ILLIMETRE

WAVES BY A RELATIVISTIC ELECTRON BEAM §3.4 INTERACTION OF PLAS~A WAVES §3.5HARMONIC ANALYSIS

CONCLUSION

APPENDIX I

APPENDIX II APPENDIX III AFPENDIY IV APPENDIX v

REFERENCES

45

47 48 48

49

A1

P.2

AJ A4 A5

R1-RJ

ABSTRACT

This paper deals with the study of electromagnetic surface waves,

henceforth referred to as surface waves.It has been divided into two

parts ( i ) A general,qualitative survey and discussion of various

surface waves, theier properties and practical applicationso

(II oJ A deeper analysis of a f ew selected surface wave phenomena in plasmaso

In the second part emphasis is placed on the interaction of partic l es

with plasmas via surface waves.In particular,critical angle phenomena

are investigated for certain types of plasmas (hitherto not considered

in the literature~ Beam-plasma Systems are also studied,and th~ ­

limitations of experiment and applications a~e brought out.

The reasons for the format and length of the paper are:-(a) -the abun­

dance and diversity of experimental and theoretical material which

mus t be included in (ilfor completeness but cannot be treated i n depth

for luck of space, (b) recent upsurge of inte rest is plasma-waveguide

propagation prompted by space research ( reentry vehicle plasma studies

and sub millimetre wave technology. )<dt he formidable background which

must be included for familiarity with principles and notation (partly

presented in app endices)

ACKNOWLEDGMENT -I am greatly indebted to DroJ.L.A.Francey fo r invaluab le guidance and discussion throughout this projectQ

SECT IOl'!-1. INTRO DUCT ION.

Surface waves are believed to play a role in events of cosmic

import ahce (inter-galactic and interstellar radiation) as well as

such intricate orocesses as that of human vision.Thus it is necessary

that a classif~a~ion of their behaviour be made. Unfortunately exact /\

math ematical solution of surface wave problems is possible only

for a few rather artificial boundary conditions (considered here)

Thus the discus s ion in section II has been confined mainly to

physical interpretation. (l eaving points of rigour to the mathematician ) fn any case it does

not appear that rigour (or lack of it) has any bearing on t he

validity of the conclusions ,which have been confirmed experimentallyo

SECTION II

§ 2 o 01 ·1VH~T IS A SURFACE WAVE ?

The electromagnetic surface wave be l ongs t o a large family (.I )

consisting of hydrod ynamic,acoustic,elastic or Rayleigh,and

seismic surface wavesjall posessing simil8r characteristicso They had all been studied or postulated by the 19th century,

and some,like the acoustic an~ elastic waves have been used

recently in transducers for optical sensors and piezoelectric (..L,)

devicesoThe concepts and principles can be illustrated by

the seismic wave

I I

h c,rd Hc.t,_1'\.._ I

~;~_ .... _ .. _.!.._' --~.:..,~=------------~--""'_'--__ - _- __ -_ .,....~---__ /

,+.

I

~·c-~ ~ Mo.\;.:.,.., o..\

._.__ ~.-.., ... p v<:..I o~

s.CJ,.,d = 3oc· ,,...,~eL- 1

i=,9. 1 - Th~ Sc•Srq".<... S01~ac..E. \t-.'a.v(..

The three waves received by the seismograph are respectively, (D

~direct wave @ -reflect eel wave, ®~surface wave o The surface wave

is always received first,because of its higher group velocity.

2 0 011 HISTORY o~ SURFA C:' WAVES AND DEVELonrnNT IN NI RELESS

TELEGRAFHY

The study of electromagnetic surfBce waves was instigated by

Marconi,s transmission of radio waves across the Atlantic.

This inexplicable event lead Zenneck to consider the possibility

of evanescent~ waves akin to those described above~which woul~ be

guided a long the earth,s surface oHe showed that in fact such waves

were consistent with Iv~axwell 's equations,and suggested that these were responsible for intercontinental transmission. Sommerfeld followed this with an evaluation of the fields due to a

vertical dipole over a plane earthoin this asymptotic solution he obtained two distinct waves,the 01dinary or spncA ~ave,and

a surface wave similar to that of Zennecko Whilst these principles

were sound,hie solution was quite wrong,for in fact a dipole does

not excite Zenneck waveso

-+ In this context, evanescent waves are those,, E and H <lecay asymptotically to zero in spatial directions away from

the guiding interfacev

~et us examine the correct solution to the electromagnetic analogue of diagram 1 .0 )

St-.

I re.,.,,...,.., - c1. ·,.,....~c....r w°'"'-' --- .) ' ..... , .... ...

::,.,,,-y +- Ll.. v' 1 -\A 2 c...i--,.2.lj.1

:l.. I . '~ . \).. ~ -. > :><.. ;;; I . ); ,c. \Q () .z. ...... 0 I v ,,c

}};-, "' --- ~~ - . r:,. 1. . · ,, ...- 1 1

(10 1)

E. 't JX,

F~ [.1 +JJ·t..,..i e.-""e. .. -+ ... c-d.:wJJ ~ •P""'- - - - ·--,--

l , I ,

w = ~ k. R ...... ,_ ( 1- (.A 2. ~ 2. +') ( I . t'/. J t: I -s.."' .A. 1-µ2~ {JJ 2, 00 1.

e...'<' f<- t>--) = ~ f e- IA. d..,._. k.. ·== :i..1v ...

I ,, , I /

I ,,,,, " I ,,

·~ Tr .,)""-- > 1'-

Fi9 • 3 !:' W = ~·Ii f _, 'J: =U:: ~ f - _r-.s;. , "''\'') .> 1_:::,, !:' c:i... r e 1_;"' \- ve.c \-ors

This result,first obta i ned by Sommerfeld emphasizes /despite its

complexity / the three waves present in such situation.Sommerfeldk error was that he took the incorrect square root of w as his

a rgument of the error function 9 and thus obtained the Zenneck waveo

Later researc hes, in particular van der Pol used the correct Riemann

surface of [W 9 to obtain a different type of surface wave . Their Jonclusion was verifi ed experimentally by Burrows /19)6/ who

t r ansmitted signals acros s Seneca Lake in NoY.state.

The Zenneck wave is a "slow'' wave with \Yrn., ...... < G , V'rhilst the correct

solution yields a "fast" wave with vl'"'-~e. '/ C .The Zenneck wave is attennated exponent i ally ,whilst the wave in 0cv(2--01hs attenuated

exponent~ally at small d istances from the dipole,and inversely at large distances. /This is the so called antenna at t enua.t ion fact or/

A final analysis of the problem by Nor ton,enables us to interpret

the mechanism of radio transmission.Norton showed that the total

Hertz vector J(' for the dipole antenna can be decomposed into three

components:- Jta.., Jtb; Jl 5 - ,epace wave, nl'forton"surface wave,and

a "trapped"surface wave.He showed that the "trapped" surface wave

is hard to detect and separate ffom the "Norton" wave,and exists

only for highly inductive surfaces /in fact it requires a surface

impedance angle 'f > 4-5"' oThis does not occur in transmissions over ' the eartb .s surface .

• ......

\

Thus at small distances one obtains mainly direct space waves, and li·ttle contribution from surface waves/unless clos e to the surface/.At large distances /out of line of sight/surface waves

are usually the dominant contribution with space waves reflected from the ionosphere.

- ---/

I

/ _,/ ...........

\ S: ~c;_ -L We \( '-_ ,, ,~ -r~~..:_ ..J~c\_

\ \

""' '-,

~he ionospheric reflection process can be evaluated easily, because •

only the space wave undergoes it~

The Sommerfeld solution has been modified to cover the spherical earth~and surface waves are largely unaf~ected by this changeo

Diffraction effects are negligible for distances less than 100 miles because ~/d.. ~ tl -.... .Similar· analysis has been carried out for

the dipole para llel to the earth's surface, and also for both oases

for the magnetic dipole.

Wa it has also carried out calculations for discontinuous surface (_4-) • -

impedance of the eartn./correeponding to land and sea./ but tnough

much more complicated, these results bring no new principleso

There has been a great deal of confus~lon in terminology especially

bet ·ween the '' ground '' waves used by engineers, to mean components corresponding to n.h, 1t1 in Norton' s solution, and surface waves such

as Zenneck's,but these differences have been clarified by Wait.

There is an important difference between the electromagnetic surfao

wave discussed above,and seismic surface waves in fig.1o This is

because whilst seismic surface waves wa.re carried essentially by

hard core,below the interface 9 e.m surface waves have mos t of their energy carried by air just above the g~ound which is guiding

them /the small amount of energy carried by the ground is due to the finite conductivity implying some kind of skin effect./

§ 2Q02 SURFACE .. 'llJA YE§ AS SJ)LU'I·IONS OF MAXVVELL 'S E.Q!JATIONS

Having shown the existence of t,he surface wave in wireless

telegraphy,we can show that surf~ce waves are particular solutions 0 f Maxwell's equations for a general electromagnetic problem!-5)

.x..

1 er~ ~~~~~~~o~l-.-, -.-,-_.~.~.-.-J~J~1~~~,.-.~,~~~~~.....s.,, ~

-t ;) , ; ·' ~ _: .... : ~ .. · .... :• .., ~· :.. . ~ ,· , D~ e..\e'-""°';c...

- . . . ... ... . .. .. \, ~· . . . ~. ,. ~ . ----- -----=--'------------ ------ ---... .., ,~, E: "' 1<..E:o

C-;:, ria vc. 1-: .-i_s H°' I.( S i"'' c e

F \3. 5. - - - .-. ~(:t) ~ (:x-c{} --:/~-::."\~ 0

.s:..,c.e- I hes C1 _':::) C...vl'Vl.pDvi<i::-nl' 0¥11),, K-.e.- v<?c ~or pu\-ev-,(-;o,.\ A ~o,r 1~~,S

Gov-~1~uro-h0v1 w;H he1.ve:. c1 ~ C--01'¥'povie"'\ o"'\-'.)· -= '\; ()c 1 t) (rv •no\ep1.2,\dev,(-

o( 'j ') 1-h H--;s '-0 . .S.<2 Mc.;x1,,vell S. eqJethu.,-,5 v.;,-.~~Jc.e \-o

0 2.t ·i:J~:c .,_ 21 :L-+' d X-2..

R. =

-;-d Lo/ a~.,_

-() ""'t-' -+- ob.,_

J, t. - ,k CA-t --kt

J L + ,,{,, <-~t -kt 4

w he.v-Q.. .A.._ = ~2. +

e_,·'1.. = >,> !-

T

;,<.. <- 0

J

.!.. K. Kc - p\-C f>C':_c) C\. V- ·, 0 I'\

\(, 'O

J.,

pre pL- :;o. 1-; on

The integrand has poles when the demominator of R is· zero .

(206)

(204)

;.e. . i L-= -,h e-et-kt • It also has branch cuts which separate the r egions

of integration int 0 3eparate Hiamann surfaeesc t' is evaluated by ms~

using saddle point ( s t eepest descent ) method of integrati0n 9making sure that only one Riemann sheet i~ used ( to avoid Sommerfeld's

mistakes). In general t' = 5(~01~ f,,-ee_ .-e::-3'c"">) -..... 2~~ (Residues.)

Note:- In certain unus ual situati0ns the methed ©f steepest

descents is not valid,and one must resor t cumbersome asymptot ic

expansions .

i

, I / 1

' I '1 • I

. J ·~ ,,

•

! \ , ... \ f ~

\ f;

' ,,,

I

\ t~ '

I '

I

•

Typically i:A.JJ-~ are mapp6d conforrnally onto c-_, J-YL and the p:roper

Fiemann surface is shadedoThe contour of integration is deformed

lnto S. D.t, and some poles ar·e c.rossed. 'I'he poles,corr·espond i ~1g

to ~i~ -=- oL P + J- ~ r can be ·v i e wed as lossy \~1a1res with

attenuation e.·-«p~ known as ~ E1aky v.Tave modPSo (Thes e wadf's rnust be

included in any reallis tic radiation field~1 and can be used to

interpret Cer·enkov radiatioh11

and aAsociated effects ) .; In the ca.se

cL ~ = o \Ve have c.. lossless \¥ave of the f'or·m E o<. e..~r l+j ya..-x.. f:5~ 1r ....-G...1:)) which is evanescent f~om the interfaceoThis is the electromagnetic

surface wave.(The fact that it is lossless arises from the assumption

that the conducting surface has infinite conductivity - in physical cases ohmic losses will occur,and slight attenuation will ensu~e) • Its existence in a particular system depends on whether the surface

§ 2~03 MORE COMMON 'nYFES O J_ F s IB1:B1ACE vvAVES

The surface wave derived b . . 1 I . a ove is ideal,propagating without radiation oss~ n real physical systems deviations from thi ·11

Pert b t· - - s wi occur and ur a ion techniques must be used but the . .. 1 .'

uncha ~d . 9 nr1nc1p es remain . ng~ oHaving proved the existence of ·the su:r·face \F./ave ( th .

other si more efficient method f . . . ) ' , ere are ,... b . - . . . s o exiting 1 t " 11ve can i1ovv study its waves cac with its TE or TM ~ ( 9

1 - & moues or sometimes for asymetrical structures

comp ex non transverse and hybrid modes ) o si

lCl) 2 e. V1 Vl t::.<:.. k "'1a .._.,, - 1') \ '

~ ' ~ r Vt Vl e.... "'- c~ v L 5-...... ' c\ E:.c. ~ -~ ..:.i C- p ! Cc 'i (.j I~ H € c-t' ..:v~' Q <'-

1.,~L ~ J'I :> ) GI.:= (:; .J l)'l.. :. .....

E E

'"' I i ..

/)' I ~ If\ ·--=-+----, r -c.

(b)

f:. < -

'

(b) Ro.cJ I 0 \ G ::J \ i nc1 \' I c..a \ .S:uv-rc-.. Le.. w c ... 1/e.

11 H

0 (: (X. I= I' t-. l-

Y-Q'.., c.:J-, .(~4 \ ~-- ;;.~~·\ ] .. f/f Ill

( u. )

Ai lc• )( •.s cf >.;;)•"''""'.Zr\"'~}

\

&, I ' (.>

1 \ /

/

.. v

- -<::. n ( G) t? I V'\ ,,,

( 1 <!.~1srh t:'f -:::: ~~ \) l.'I... )

(' (t:..J ..:>o i'V) . ,~.....-.\.z\d - &00 bav t .. J'o..v<.... <;,, Vi).0~ I,, '-l' .' ) 1 , •::, - · II""" -JC, (. e...

r

I

I

( •

l S ' cJ e. e.. \ e v c .. \· ; • .;. ,'\

·J ' " I

/

'\

/

\

I

' I

q ~ _ ,_ r S · · ( ';:;> "-.... Wu ,,'- ) (bl \::- "-..... e I ~, c. t· , ... ""

The solution to the Sommerfeld GoubaV. problem is obtained by use of

Stratton' s theorem which st9tes that w1·+hi· n any v - is otropic homogeneous

medium,every electromagnetic field can be represented as a super-

position .of two e leme.r1tar·y wave :functions -

+ JY'l 0 ., (_ - :rd(_ J,~ -;;...Jt, tr\ -l-., k = e v "'-' .J K 7.. - J1,-<- v· ) e r:..; (:L OS)

,I _ _ ·!3"'e H L1'( 1 :2.. / .,_ ) .:!:,' ( k ~ - c._.1·( , I ' - c. \J '' - "' ~ c.- °' nA,K ~

(\ )

\-1 t= ( \ ) -= JP ( f ) -r j rvf> ( f ) ( l-10\-\\.,e.1 \.;, .... ci/10\1'1) , Vs..;,_. I~'=' c "1e

l viO e c..l e r::>:>V\c..A E11U!..). Fcv- \ hcse.

t::,~ \I I e"

( i -4- I . _ - . _ _1 ~t ,.. - - - -There exi st a variety of other configurations e.goelliptic waves,

waves guided by ribbons ( ~1hich are in fact equivalent to cylinders

of r·adius t the 'Nidtb of the ribbon t~~. In fa.ct a.11 possible wave

guide shapes can be examined by conformal mapping onto the unit circle, but these do not intTonuce any new principles ( They are

equivalent to replacing the radial Bessel functions by ~thieu's or l\1acdonald' s ) . J\ ny\¥ay

9a 11 regular shapes of guding Au.rface

provide a certain definite surface impedance,and the wave propagation . , d l tO)

is independent of trJ.e vfay that irr1peda.nce is proauce •

For a complex wave can be s hown that it is equivalent to a suitable

unique sum of plane waves~ano total i~pedance is the sum of those for individual waves.

Equations of propagation for a Zenne ck wave "L~-=- l~s ~ j. X~ = [ E" .,.. .. l ( i=. 'n. 1-tc.- '""'f>'c.(c,..,;:.~ '--o.,d~r;oV\)

H3,.. j ';J"O

)

J ("'+-)

o~ \·s,~\e.,. \-~ 5 .... :1 cf1v13 $1.H"f "(e_ > J'~Z '>C"J \v i·;Oh IS

H h -== A e,~ r [ c~ {.-..' - ,(_ ,, ) x - c j r' + P>,, ) $ +- .i .:.. ... t J (o~t-c.,... ~;e.ld.5 ._~,, \o.:. d1.2C\'"'ce.d fvolM- LI) -" ~) c..bwll.2. ')

$__, _1 :>'-; \·.J\e I I \ I f_ ' !'>' 1.-viv'-'A'-'""c::..c c...oni::,, ·,ov· ~ 01nrc.'' "" 1-:J

z - £L -

Thv.s

H -5

. ,, -~ - K"' X:;.

1< or· a n ort.inary conducting met al Z. ~ ~ (1..,.. j) ( k0 /2 s-Z..,) i:.: 'Ihus such

(_2 Ol J

a surface is capacitive,and will support a T~ surface wave with X 5 = R ~

(both small) .In most practical c ases~such a surface is not used

because X~small irrplies h" sma l l ie a loosely bound wave.

(~" will be small - implying small attenuation along direction of propagation ) •

J. good surface wavE transmi~sion line is obtained by increasing

h" ie the capacitive part of the surfa ce oThis can be obt a ined

by coating the guide with a thin layer of dielectric,or corrugating it.

The s oluticn to the corrugated guide is an interesting one employing t ,,,

the use of 'Ploquet's t h eorem,and applying the appropriate boundary

conditions. ~he wave can propagate across the corrugations,o~ even obliquely.

2 .032 POLA R~ZA TION

So far we have considered TE, and TM waves onlyoin fact,e surface wave wi~h an arbitrary direc t ion of polarization can be transmitted

by a diel&tric coated,corrugated guide. ( see fig.10)o

0 D·, "~.\-,ov,.

u~ P""P'".'.:>nl'•"'"' Vl~h:d .sepr1.'-{_ D; e ' e.0\-Y-I c.- 1--\0..!> · ~.-v-c-.r .. 0\,s.

l'O 0.CLC ~oclo.. \re.

f=" ;_s •O. (_I;$)

This configuration ha.s been studied by Hansen,who showed that it is

capable of supporting TE, and TM waves siwultaneously,but of course

with different propagation coefficientso(usually resulting in one

mode being much more dominantjo In this case the TE mode has its electric field across the waves~ corresponding to a TEM mode in parallel plate transmisssion lineso

For small plate thickness,the plates have negligible effect ant the

TE mode is governed by the dielectric slaboThe TM wave has

parallel electric field components,whieh may be shorted out in the

plate region provided the septa are close togetheroThus the effectivt

ttickness of dielectric for TM waves is that above the t::eptac

By alter-ing both thickness an:i characteristics of dielectric we may

produce both TE, T:V: waves in arbitrary proportions --/ arbitrary

polarization of surface wave. In fact this is an approximation,

because higher order modes (we have only considered lowest mode)

may be present between corrugations because the plates are not

close enough to short them outQ this only increases the complexity

of the device,but in practical systems,one can show by ingen~ ous construction

9that the polarization can in feet be ~ltered at willo

this is useful in antenna design.

2 . OJJ ASntl\E':5-IC 1•:;; v:;;s

J~n interesting case of asymmetrical ra.dic:;il wa>res travelling (;,,<;)

on a flat surface also been investigated (so far in this paper

only symmetric 111oces have be<:n considered.") For a Hertz potential

JL with a single component in the } dir ection we can solve

l\f ·118 t• ,,:exwe equa ione. C)"t..:r\.. 1 0:h: I 6;._~ 9""".rc + \<. '-.3L = 0 - + - ..,....- -- -1-JY '- I"" oY- 1-"' Cie2,.. o~/-

Sc.,pc-.'f"0.1···~'5 ,.:C<Y.IO..ble~ C\1-\cA o.\( OW.•Y'\5 vC·-no-l·;o..., w11'i-.. e v-J.?- 'C:lbro."1"

("-" - [ ll) ( .., (;z.) - \A, J l - A H"""' f-;, .- ) +- b 1-\ ~ ( r, .--) J .,, ~ -:: ; ,_ (~ c.'>) , · _ 1 1 r \" ·~ I "'- ~-·~ "'"" ~ w~'(_N\. _:j'e\o.s I-he. '""'e.ld """r"'". •, ,,_ D ('..(.) -<A<'. -.,

. -1 =- ~ lAJ I:' c vv o H ( ~, ) " r e .) , 1 ..- '"" 1- e. (..8-1 \"'· ' r .

- ,. ) -U.?,- . )~~ ~e-<=- -(A.,; rs f-\~:i.)(~.)e-''";» C.c-;,L"'~) ) H&- =- -dc~· ~clJ 'sH: (~.--)e 5,,..,_(nt:'-J,

- - -;i. (=<-) ( - u:~ l=; - 0 I~\-\"'- P") e 5;_~"-(_vi<t:) 1 ) ::: 0

c / ~ ~

~ 3 (;uc~ ... -i.l;, 1::: .... -= - u..- is 1 H \:. ,. ) e , · / e ) · l"" l"v ~L V\- \.....'"'

From above 21 = Z.2. = ~f-/tJe:;.,. independent of the order 0f the mode~

in sharp contrast to the axial wave on cyliindrical structures.

hlso these waves are all pure no hybrids and a novel type of waveguide may be designed to transmit them .

Wwe;:... Sl110-p:z.d (.,.0 , de. i=;;,,,~ As.;;i•"''"'"e.t-.--~ c \.v'ovs. T....-o.v.~1""; ~s.( C.\I\ .

F;9. \I .

§ 2.04 F H' .. . ') fC1\l, IN1' RFRFTP.TION OF THE ZENl'rr:.cK W1\ VE

Consider a vertically polarized wave incident on a flat surface . th .&:". • t d t . . (l '>) wi ~ini e co~ uc ivity v,

J

/(//! //Ill ftll

i::: ;.'.:: .. \~.

f he reflection coefficiEnt

'1J ;s ~ fSv-e..v-Js.\ e.1 a."'~\e.

l{--'B.-ews·"'-'~ " = '}'o - J X:

f'

\vi e.,d ' 0 1'11\ @ fa:t.""' }Ao E.-.. "' &., ei-l..= c

is determined

C ,.,I · ~· ~ (r: <,-. 1) ,·~ (."t,--p\e" .. ·'"''-'~ .,., '"'':::.. _.,,.. . .,,._ ...,.,, ,....

I.~ K..: O J ~e... wa.vQ. 1S Vio\IV\o:j12~"'e;:;v.s; ( e9.._..-.. - 1~ho-..se. ~ .... , ,fc~c.e:::.

G.o 'tv1C; de. w; 1-h .e.-9 vi - Cl t.v-. p \', 1-'vd e .Sv,.,.. fa<.. e.s , c- .-- o. v-'2'.- pv-..-o. {( e. I

ro ~eVV))

(:::uo)

However if ~is finite,we can show that a homogeneous wave cannot

satisfy f=O .In fact we can derive the relation between the

equiphase and equi-amplitude surfaces for the inhomogeneous wave

which does satisfy <('-:co • ~'rom fig .12 we- h '"''"<:!.;

v-J~e.r~ 5~ vi'/--- .=. ~·"' ~· 0 loS..~1 'X ·-Jee$ r,::. 5>~,..-h. 'X, (..-05.f =: C-o5').-'0

c:.osh. Y >-js""''}-10 5,..,h X.

S, 1-1 = A ,-, -jpcd.. (. fl r---: ) - ~JL - IA i.. ';} I VIC~ b:L <c.,- e ;; G.J -,J fl<> Go we... Vi C'-.ve. H h '.L. == A.;;_

vJ\..il.v-!.. Y~ fo G.c-:::.1 ·~1cS~wi-..X + 0'p.0 s'iVl');o~·X. _; (..(.:i..""'(>cs~"'i-10 s--:.,-lX-ipou::t;'f'occsh.X

But this is the equation fo:r 8. z~ nneck wave, provided Gle (l-l::t) >o ~hus,a Zenneck wave is simply an inhomogeneous wave incident on a

surface at a complex Brewster angle. Fqui-phase surfaces ~.::.~""''\Jo - jccst'" = C<>V\SY.

Equi-ampl itude surfaces - :>....cosy~·+-~[«" 'Vo== c..:cv..s r. Hence these are mutually orthogonal lsee fig . 13).

Tower transfe r a.ngle = '\Ju ;::, ~(~' )

Thusjthe ?oynting vector is tilted into the surface to account for ohmic losses within. We can obtain a physical interpretation of the process by transformi ng coordinates :X.. I = ;L:::.v ' ~o - _'.:)C<>So/c. 0 ! :; ::::!>'"" 't'c:. -..- X 0o,;: {i.,,

.', H 75 ·::: A 0Xp [-<) ~o ::x'.-1 Go..> k- 'X. ·+- ~a ZJ' ~ ·,.,, ,k X.] which represents a wave travelling along the x' direction without attenuationo see fig. 14

. \t' /~ ~· 'f

I Z e.V1Vl~ k W C)\../<I-

,, , II ; V1h> ,<;, , ,, \<., .-,~ c;..

I f i9 \4-,

2o41 PO\ EE C./' F.r: r ·.; ... B ~ J\ ZENNECK 'N;\VE

The power carried by the wave is u

p ·-= .L <fu:..,. S C E \-\ ""'"d .._. 1 = .J... .fi_ 2"' A A.*' ~ - 2 0 ?r ::1.,, L J .j ~ .__,, .) 4- "'- ,, K 0

Its phase velocity is determined by ~(p)·= (?:> ' '1.Y = .i:d__ ~ .<:::::'..... (I+ X. i. - R. 2.. )·y,_

P ('.':> ' Ko s. ..,, () .. 14- )

It cc;,n be great er or smaller than c.. depending on the surface.

In fact the presence of surface waves in a transmitted signal can

often be detected by their arrival time which in general is

different from ordinary radiation. Hm·.rever this is not very convincing

for dispers i ve media or signals with great frequency spread.

2.042 NOT~ ON OTHER WAVES:AND SURFACE WAVES IN DIELECTRICS

The above analysis can be applied to the Sommerfeld Goubau wave

[and the axial cylindrical wave) , and although more complicated

functions (Hankel F~ ~ )must be used the only new result is the mixed mode propagation of the S.G. waveo It is found that the

circular cylinder can support cylli: drically symmetric T l-\ c"" ; TEc,,.._,

modes.as well as hybride,all of whic.,h have a low frequency cutoff

except the HE hybridoThis is the mode that can be used irl

dielectric image lineso The HE 11 mode is carried on a dielectric rod

(Not much different from the ordinary cylindrical conductor) ,and

thus it must heve a plane of symmetry containing the axis of the rod.

fhus image plane tgood conductor) can be used to r6place half the 04-)

!'Od and the surrounding space~ -he image g1 ide is more efficient,

because fer tightly bound waves the attenua t ion coefficient of the

line iA swaller than for pure dielectric.Per loosely bound waves

( rarely used in pr a ctice) this is not true,because then the dielectric loss is small. I t is easy to resonete the HF mode by

It

placing two plates at the ends of a section of the line,and numerous experiments can be ca.r1-.ieci. out to invest igat € the propP.rt ies of this unusual moieQ

12-

( c•'• d '" I ·, "' '.:> :-, \~· ... 1

I \ ~ I

D', ~let:. I· ;,(

Dielectric el3bs and fibres h8ve also been studied.

Surfa.c e waves in die le ct rics may be v isuallis ed as being ''trapped" i ns!de ,undergoing total internal re f lections at the boundaries.

(in addition leaky ~eve wades and continuous rqdintion spPctra are at each of these "reflections",)

/

(',> {'~ i -? ...... "' ~, F t<j· 1£. V\it'.v<'.. ~\'<c.[:>p~0\.t i.n (>, c-A',.-:\Hh •L

Since the wave usually penetrates the whole diel~ctric it is

customary to employ such frequencies and dimensions to reduce

the energy flow inside it.A single fibre dielectric guide has been

used succesfully for visible optical frequencies,emplosing

the HE 11 mode with srr::=il l ro.dius which satisfies above. conditionso

Because of this, at t enua.t ion wo.s very low ( .2100..1::/"-•"')t he losses being

due m~inly to rPsonance absorption o 9nd i~purity scattering.For low

dispersion ;power handling must be limited to 100 mW.This line can

be used to transmit gui~ed laser signal so

§ 2.05 EXCITA ~ IGN OP ,::: u r:.FA CE WAVES

20051 FIE.:L. D ORTHOGONALITY RELATIONS

In crder to e±amine the excitatio~ of radio surface waveP 9we must

considPr their field arthogonality fe lations.Any guided wave is

usually overshae vwed by the supplementary fields which a re generated

to preserve continu ity near the source and receive~ By considering

a decomposition of the total field into a sum of these two

componentsyand applying the Lorentz reciprocity theorem we obtain

a very poweFful formula yielding t he amplitudes of ~urface wave

modes excited by any particttlar configuration. See Appendix I

This elegant result is obt aine~ ~ ithout resorting to solving

the equations f or t he total fields.

2.052 EFFICI~NC'l

13

The abov e formulae may be us~d t o predict which launching circuit

will ~roduce surface 1vave energy vrith ~axirnul11 efficier1cy.(rl ==(Power carried by surface waveM_power transmitted by· launcher,b)If . t he launcher

is of the s a me form as th e receiver '1 <'.. """'""'.w.'J = 'Yl r~~~0 ,.._

The launcher considered at the beginning of this paper is rather

inefficient and a lot of work has been done to improve ito

For a line current s ource,t he optimum height above the conduct i ng

plane (without the dielectric) , has been obtained in a very s i mple lf.!.) fe s hion using a relatively forgotten theorem of Newstein and Luryeo

Th i s states that if the field in the region of the line source is put in t he fern: Hj ( v-,e-) == 1-<0

cK...- ) r- f(v-, & ) . [w Yie.-e. '{{v-,9- ) regular

in neighbourhood of -r = O -> c,...,J H~1 -Hankel function of the 2nd kind 1

a standard solution to the radi¢al problemJ~then the power

in the field is proportional to the rea l part of the field

at the source. F\ -=- ~E.G [ I -r 6t ~ l ~ ( o , D )) ] • ( This can be

shown by obt'a ining the Poy -... i- · "':J vector and integrating around

the source,finally taking the limit ,-- -4 0 )

F'o r a source at height h above the plane o{ 11•).-.'V)r~h.z.!:!d ·, 1,._,peJ.c. V)ce.

Z- = j ( /~-=>··· ) 'f ,_ X > v-1,z 1,--,c. "<:.. '"'- ( 1.. I 5) . l (l.J . 1- ) 'rl<. ... >(_. I -) - 2'i _\. l'--X>Gx,.., \_-- \....Xx.] ;-1-1 :1 (-:;. .. , ~) = d" H ~ U-.....- +- c _ ,.. , ,

(;ii. .· <..:u -r \ - • x. 1 r . . .l..)} ~ :\ -r- J ~ r> l k. .X ~ ) Ht> (h. R- ) ~ ~ .., . .:4- .-r,1 .._ ..... ..... '<-xv l- K .:.: h_, e..x.c t_:-1.: .><. :\'._ - J t<..U+ .x l }~

·- t'O (\ + J(..)

The last term is the surface wave pa rt,and thus surface wave

power f raction can be calculated Maximum efficiency is obtained

f 0 r .J, ( .<. k ~ ...., ) ..,.- X J"' ( l_ K. -l... ,.._ ) = X (~:?.. I b J

Thi s simple result d isagrees with all previous theories which

predic ted k. x. {..)",., ~ l

-

2. 05 3 A PERI' tJTtE LA IJI\'.CHf;5' S

Besides line sources,magnetic dipoles etc,there exist a totally

differ"ent cla.ss of la11ncl1€T'S which cannot be e.nalyzed s0 simply Q

These are aperture type Launchers,where the electric field is impressed upon tr.1.e apertu1·e ..

~

- .

'~ \ Cl I/ _j • I '

'l1l1ere exie t t \fliO n1etb ods of o bt a.in ing th~ so 11lt ion to such a pr·o ble

due to ~ulle·n,@due to Booker ... Cle_rumovv,both exact and yieldir1g

indentical ~esults.No pure surface waves can be launched this way, because such a wave requires an infinite front,and the laucher

dimensions are finite.~evertheless good efficiency of exc itation

can be obtained . The slot is eqt1ivalent to a filarr.entary m3gnetic m~-(['­

currenL ,Both, Cullen's and Booker - Clemmow methods deal with

Fourier analysis of thB fields generated by the equivalent curr·ent and imposing sensible continuity and radiations on the fields.

In the Booker - Clemmo\~' solution, the resultR.nt fields are decompoE

into plane wa.ves wr1ich are given the''Zenneck wave tr·eatn1ent ~, a.11d

later superimposed to obt~in t he fina l reeult.The exper·imental

resultss.gree ·we ll vJith theory and ;>1 ield. over 90% efficier1cy

for cptimum s ized apertures.A variety of systems ma.y be used

. j

(various shapes of apertt1res ov·Er plane , cylindrical or~ dielectri 13ed

coated or uncoatej corrugated surfaces) but all yield essentially

consistent re8ults.

) Qr= II fI'T.f/.i., -r,,-'-" ...l'"t - 1:. J.J .t' '.LAl~ED HOR!'T fl I •

i' pr t · • ' - l -} PFf.J\ CT ICJ1 L LP. Ul\J c_; H ~R

I a,c J.cal launcher for the sin~le wire 2·

wave carried on a thin fl- i.9 I 8. °J Tr1is V.'or·ks

dielectric on the principle

incr·easin~ outPr· .

2ne coated WiT-'e

o.f a. coaxial

Somm f er e ld - Gou bat~ is tho fl d h .... . are i o::bn

line with a gradually ~ - raCL.l.UB.

\ " ' ' I

• I V

' . I ' I I I ' I t 'f lJ'

• ' I . , I I \

I I p

.....

\ ..

- · i-. I t> I 13 . I>

- --

T\_ rl) I C.·\~O~V ~\ '''2... - . ~ "· -c' e

(_\ I ' \ ·· L (

_:)c,c-,r \ S, G

' C\ lj l'I c.t1 (! •

,. . - ::.. i VI~\ e;_

E.N

- -

•

• - ,._,. - -

~he field between the two conductors can be expressed as a sum

of IIankel functions and something similar to Bessel functions.

15'

As x- increases the latter tend to zero and t.b.e dominant mode becomes

(asymptotically ) a Hankel function,which in fact is a superposition

of Somrrerfeld Goubau modes. ThuR the ~ave launched will hRve

essentially an exponential decay away from the wire 9 and can be used

to transmit microwaves along ~ single lineoTV signals have been

transmitted this way in ~.IacI'onald rass, ::'.ontana (USA) a!li ~"aloyaros la­

vets and Obnisk ( r;c_;~:R). _An important aspect of surf ace wave launching

is the critical relationship between the impedance and bandwith

(analogous to gain-bandwidth product for electronic amplifiers,

but far more complicated) • It is found that a fraction of the total

energy will alway3 ~c stored in the launcher and receiver,decreaeing

the overall efficiency. "'or :n9xirrum po1Mer transmission, the launcher,

guide and rece iver must be i mpedance m~tched.

§ ;..'.. 06 EFFJ~CT OF' DIS COI-~1:'"'IIJ PIT IES

In any practical surface wavP tr2ne mission systE.rr:,guide losses may be

attributed to discontinuitiee such as bends,changes in size or

thickness or ~urface imped8nce,and obstacles.Cen~rally changes in

shape or thickness are the least importPnt especially for the single

wire line because it has been shown that reasonable devietions from

the circular cress s ection do not ess Entially alter the wave. Radiation loss at a discontinuity of reactance.

Cj-

~~ i'c:.1- i ... ...xzJ w_-...v;z,

~------·

..Ifl-.., ... ~" y)..- ~~v ~ {e-. <i? ~""~

e - -> ~------------"--'------------?~

F, S· 1'1

5;._. ,-{c~c<.. Wetv.; n. \ C\ Keo,c h-,._.,r e D;s.c<:;v-.,1· : ,-,J,\.j . "¢

H1= 12,~r(-\-\,~ -j\~1.,._) 4- )_<XJP('-$ ) ILJ><v~-js_,.~ -j'(J-J~~ ,_ __ $·i-J d.s. (:2..17)

E')...-= ~\;L.' <:!-xp ( -u.,'j -J(s, x.)- ~G s~ .. \°''(-s.)J1<..,,.~--e;"' ~~ L-J·Sy_-j'jJk,:--~LJd.~ w ~.. t. ·-<>"'

-,~ 001.-\.\ e<\.-uC\.Y" OV1

1"he first terrr. represents the incident surface wave whilst the st."cond ,,

describes the auxiliary field associ8ted with euch a wave - a sup~r-

pos it ion of plane waves with differe~t propagation coefficientso

Boundary conditions

S06.s\ tb..>hv-.5 \-ov- ~..,..., \...\ ~, we ulol-c.",,.., · ·oQ

t~ f'(~.) ( J k. .. : -- ~ -t- ~ \..\., ) ~? ( - ,.\ c; ~ .... ) d $ : 0

r"' p ( ~ ) ( h:c !. - ~ i + '\ u_ "L ) ~ E' ( - d ~ -.... ) d_ "<; = J C.A. l - '-t ~ ) e/X f ( -d p I ')(, ) -·<J

:.·hese eq·.i.".>t ions may be solved f or ~(~) by t he 1·;'iener - Hopf method,

which relies on Tou~ier tr~nsforms, and summat i on of residues

( far too long to be included). After substituting for ~~)in the

fie l d equations, qnd using t he mPthod o f steepest descents,we arrive at the a.symptotic value of the radiation field

H1'(_.-c..J,o . .'rcd) =: c:t~\.-.u )'I,_::; . ..., e p (K..,C..o;.,&) ~?( -}kc \r -j ;t/_+)

·::- c-11 :~ )'1

1.. s,,,._ e P( \::oC.CS fr) H ~) ( j k ol'"")

( f' ov- v- ':::> '> f-0

)

~e may evaluate t he surface wavP fielde from the resid ues at the

poles of P(>) .A ft er laborious calcula tions we cen deduce tte fra c tion of power radiated to be

p "1,-~. ~" (' \\ t.\

Several ingenious wRys of minimizing t his quantity in prActical (.Iv ) systems are des cribed by 0 9rlowand ~rown.

Wh i lst the Wiener Hopf method is r igorous, it is extremely awkwar d

to use in problems with complex boundary conditions.A nove l method - of overcoming this di:f iculty is a perturbaticn theoretical

( ,7) approach analogous to it s quantum counterpart (wi th the axial coordinate replacing t irne ) •

l<'or systems \vhere both the perturbed And unperturbed states o'Jey e igenvalue equations,ti~e inde pe ndent a pproach may be used.

I n elect romagnetic theory propagation along axially uniform structure

is described by an eigenvalue equation in the axial coordinatet

the eigenvalue being the propagat ion constant. Considez eigenvalue

equa t i on (Ao -r A)~-= -~J>s ~ Where 'f' is an n .. vect 0r -fun ct i o.'1 of i and an arbitrary number of ot her coordinates. A and Aeare linear

partial differ·ential l"-- x I'\... rr'3.triY 0perators(not containing

d eriva t ives with respect to ~ .) A.., is independent of ~ .. and is such

that the i~pe_rtu_:be d equa.t io~ A~~\ -== ~~ q'.:0

= - j0{'c has known solu:ions 2 ~ =±P '2-xe(-J0b)}, v--~<2-1-Q., ti t i'3 a complete

? s . otbonormal set of basis functions (indeyfpendent of 1J )of 8. 11ilbert

space (or unitary space if p is a discrete finite variab le.)

remain complete 'I'he perturbation metrix A must be Puch l p P ~

for the rPgion conf"idered in the perturbed ccise (.and must be small compared with A0 ).ftll changes in boundary conditions when transfering from unperturbed to pertu~bed situation must be

incorporated in ~.Tor simplicity assume single mode propagation

superposition ) "Concic er the direction in a TE

""\..

( PJultimodc results may he obtained b;y

Consider a ~E mcde propagating along mode pr0pagating along the ~ dire c t ion

(w:r~ l"O ::I de.p-2-V1ctev1c.~J in a rectangular waveguideo

x.

~· csr----<Sc;.E. -/!.c -~ For this case ~axwell's equations reduce to

0 ~}-lo f:~ t' ':I

::: 0 ___ I_ ?J,_ +,We ::: co JW }J,~ O :L'~ d c ' 0 Hx H.x..

(2 i I)

- '1J . ..L...._

;r .z . .:.k .:A V\_:r.J t a,v- WCI" e..~.,, ,d <G VV'\<l d ~ s vV' 1-~ P.n. = J ll.,_ 1.:::,, - cY\: ) L ( V) > 0.) K 2.::: u.:/-}A 0 <::: <.>)

If a dielectric obstacle of p~:rrrittivity E is introduced between

..:i:_. ( ~)) o.v-d .xi<..~) , thP pert urba.t ion ma tr ix cr-i,n be obtained by

subtraction [O 0J~ . · l~ ,.,.,. i-1.ne.. A-:::. j\..utc[<=(.,;} ) - t: ,J~(::x..-~,) .-h,(-x -::...~) ~.-,,\· "'>~.Z...f>

l 0 - \u..,<.-h ori

For li=-1 considP1· the perturbed .solution :±1 ='±'::.ex;>{ j((S,,~+-'f0 ()>.1 -t- ~

where ~oll) is the non-linear phase progression function,essential

for electromagnetic tteory,but oft2n neglectel in quantum mechanic s .

~-:::: L. l.Lp(1,) e..xp(a(1., ~) 'f" (a superposition ')-f' all but !~ "" '"' u f>

the incident rrode ). t:.C..1) c3.n be complex to accou.rt f'or cl:Jenges in amplitude and ~base.Substituting into perturbation equation

(A+ j :~) ~ e.xpL-iC00 ~~ Jl\>)1 = I_ (c-:i~i' -LlpA) f P e.;q{-ifr2i) (22:z.) o 0 V J f> ~a o

Befine scalar pr·oduct as matrix multipl ication of liermitian adjoints

(followed by i ntegration over the region of completeness and orthogomi.lity) ·;- _ e 'j. ~ o , ~ ~ - l -1 ,,,., !'> ~ v J s , "" ( e ~... )

< \T1J I \.lJ ,, = r V- r -I ~ J [-Wrr: J S~v'l.(p'\\)L) s\ .... {[''" ,L ) d. )L = ~l:::>n C-J~o°'-:r, .I,..,/ )() ~ P.1"' 0- 0- •::.

a ~<l < ~ / I A I 'fl~) ~ re -1 ~"~po ] G : ] [~rt°] s•-(e;,'')''" o¥-'\i(>-,,)<l~ \N\"'<Zre... A - [

0 a \

- ca ( ),, ~) a J

-

c\ vi cl -\-a -·N'I · ' v\ 5 t ..._..; o su l a

- ;-'"t- <~o\!\\fo'/

r. ad. 1.._,ci.- 5 0 f ec-v

-J ( r~ c ~ + ~c ) e :=-2_

<~?Ji-I fo > a \o'i d

~fcv-t-lAl~o>e < f~~ ( f q,>

s ... )b~, "~ , .. \.._! \ \\A~ fo'" ,.

~':> "') L.Lf> •

lLJ ' ._ .... ( ct{ ~) (I_ -k) d~(} 2_ - - .C <P •

cl~ - 4. "?

K..=.2.. 0

~

- < f ~ -t \ A \ ~l'- -I>> <f.,1 I fp >

' (_ L -1 )

u..p

f>.,,;-o

- -0.. "" d "-1.J w.e -CV

< ~oi-1 A I ~o '> <~~t- t 't 0 '>

,1t1 1-"' < 'l: cv A ±' ti / -

•

(~J4-)

. ~ n iter·at ion proced:ir'~ can tl1.US be establighed to evaluate {<> c.~ol l,A..P

to any orde r 5 but integ1 .. at ions, end espec ie.lly summa·t ion;:i c)ve1"' stat es

b e com. E p ~c C) 1-l. i b i t iv e for \'\_., ;;> L • 'I' o i 11 us t rat e th is .me t hod c on s id e r

a ridged dieiecti-·tc Ptr·tlct u_rP l discussed q11allta.tively befo:r·e).

~ I - - )1->,i b -1<:--

~-

/ _,._, --~'> ~

~/ /////11////./fl/ /{/ I( fl//f((f/ flt ff '(I {fr' tf{1. tf/{f,, ''' 111 f ·'f' If:· (f't'ftf(t//l'•/,t'r1

R\d9~\ d.\e \Q.t., \1 rt.. s. ... <"'""~"Li.;.. C'•" c, :_..c, d-1.)ck,, :) P'~1t1-... F10 .:ti.

\ itbout the perturbi~E ri6ges the structLlre s upports R se t of

discreta s~rface wave modeg Rnd a contin~~m of radiatine modes. The pe1·turbations qct ~s ~ cuu~l~ng between these , and ~ence our

previous equat ions n1ust t~ modified t o include the radiation f iel1 ~

Consid er the c ornnlete set of modes ..L

-t ·= -o

-

-~.Wf"'9

~..:;

- I

1- ; -c ) r Go I )\...

-·~II - -V'I Of'I <;;. ' e_

-"1:10 is incident su1:·face \r..1ave, ~- reflected, ~(\~) conti11uous r8diation

modes for a.11 ~2 < I<.<.. • 1I1h e per·t1.:i.rbation rr1atrix is the sg,me as /..\·i er' ived

previOlJel~; ,v.: ith 2)1..., ~o .. x~-== °{"Hor· the ra.diation field i;Vh er·evr::r eum1nations --- -- -- •

over p are evalua.t ed

· 1 d th L ba trRns~ormed into integrals and invo ve , ~ese mueu ~

by saddle point method ) •

c,,

l 0 h 8 11 ~ b

t= .5 :2_.(.

c. ;:::: <)

p E c:A ~v-,?... p ~ ol p ;. d ·- -b..; VI.. p 0 ct = e

5

., ....

'-"'"C\l>v- cf I-Vie_ .\JV1cJ-,o·" a--(b)

~·s :(.3

This technique can be applied to

in the form (Ao ~A ) '±' = ~:[. and rarcvvitz that time harmonic

any ooer~tor Pquation expressible

It has been shown by Bresler propag~tion in any linear,

sourceless,tiree invariant medium bounded by open o~ impedance

bounaary conditions can be expressed in the above formoit is simp l e

to ahow that raxwell's equations (even for anisotropic meei a ) can be

transformed this way.Then we have

0 0

0 0

A -0 0

0 0

I ( . .L I ) c·-1 I ) w V)e.V"~ 2.::: 'f - J0·Jf\o ,~c.....it:., <;- 2- ... .:::, /V'-:i.. f-11

0-V\ J.. ~ .:::: [ ~)L E H H .::J J 'j :x_

~be power series typ~ of rerturbation theory ( Fayleigh - Schr~d!nger) is l~rnited in applic~tions because of ~mall convergence radius and

an artificial requirement of seperability into eigenvslue equations.

An improved modification is the Brillouin ~igner method which

construc ts eigenfunctions in t erms of t he unknown eigenvalues

(yielding continu~d f~~ctions which converge rap idly even for gross

perturbations - A not small como~red with A0).

Plthough superior mRthematically, this method offers no ne~ physical principles and hence will be omitted.

An auxiliary method which treate boundary conditions as perturbations

has rPcently been d eveloped,and has foundapplications to surfac~ wavA etructures.

A generalized rr,ethod of solvi~g complicated boundary conditions

can be achieved by intr·oducing { A""~ , a set of parameters (generalized

impedances ) which complet ely specify the ')ouudary c onditions .

For a function L-L(.:l;. ,.'J) s::it isfyinf the reduced wave equation these may ( :( I; I ( 1. I . L - - \ c. . \_ ,,. . I be wriJ.; ;pn as I\ ... , 0 Y t V :? ... < -''> "'' ,., ""'C.. o,;; p ::. ~...,'>•.: "(. ~ ';>

_/NI c· \ ~ _ 1 _g__ \ V~ L ')( '"\) == o o--L j ""o c·22b ) ..,.,,.... 6 , . -

(where mITfti~lded 8t~·ucturea are involved 9 normRl ~erivatives must

be used ) . Ti'or '.3.iscontinui t ii:>s ,a different set t _u ... } must also be

int:t·oduced ~ · ost single mode sy st ~ms ( N= 1 ) have be e~1 sol VPd: --thin

d i eletr·ic slab,corrugated surface etcQ Certain double mode (N ==2 )

problemR have also been solvFd . Tb~ method described hPlow appliee

to infinitP structures,but all prob l FTS irvolvinf finite structures,

but all problPM~ involvine finite structurPs cqn bP Aolved by suits ble superposition o The { "- ,.,, _, can be found from (r~) t hPoret ical

(b) experimental reflection coeffici entso ~or sirnp:icity consider an infinite inte:rff;.CP 11;ith R e ~"- ( perfect refl ection wit~cbange

uf phase). This ~pplies not only fer real 9 3nd dist rihuted dielectrics

but also artifici~l die l ect1ics.For real angles~we can approximate

analyt i cally (by Ghoice of !\ .... ) • Thi 8 is a finite product approxima­

tion of · '.?rder- 1'J - ,_N ~ l' ·1 - j- k.~~t: ) K :: 11,= , )\ ,,,..,'

l ( vi-~~ h.•~p,,~"' \' ,; h ..c

$:.~ ,,-~ L l.- \,._., cA V {!. _>

If Ne ~now (theoretically or experimentally) the ectual value of R 9

we can trunc:ate the sbove product at a part i cular value N-1 and

insert the correct value of ~N to account for the remainder.

R will now be convergent, but the "A's will become functions of No • . lr-') .:x; •

However 1t can be shown that 11m (A )-A is convergent. N _.,c.10 · ,_, ,.._,

Pinally,the limiting value of R becomes

Roo::: - it. (1- J ·";,c~s G) '2....x\)(jko<.o~&/}\..":) R--x1)(~CN::.O) (2~3) "ft' ( \ + j 1

'"0 u..~ ~ ) ~ p ( - ~ ko G.= e / r.,:: ) However f3~ a finil; croduct the true value of ~~ ~hould be used

because otherwise a phase error is introduce~.

Alternatively the exponential convergencP factor hae to be approximated by a product representatio~ i.e.

ll e.x~ (d't~0 Vcr-:>Cr//'\~) ~( l ·~ j ~cc.cse )N (2211) ~=1 ('; 2._-:;::;

I .,...yn • 'rhis introduces an extra multiple pole ur zer0 ( bi1t does not affect

the analysis because it uccurs far from the region of v~lidity

of the expansion.) This technique is simila r to ~iener's prediction

theory o For a ?11i table N and { 'A~j found this v•ay ~ e can construct the equ iva~ent buu~dary condi tion

~ ( ' -) ) -11 1 - - ·"'- t,\. = u I ?-(>\ '-':!

Consider a cielectric slab (.?._ "?,c1)

o~ thic~ness ~ backed by a corducting

\"' h<Z..

1-vh«'..v"'- p .... == ~o (,;;-,(;rJ

·r We cari. appro"'":imat e > "" ' -.:.. b.J t - N terms i ntheir infinite product

representation~and Drocf>ed -':o find the As. nnri hPnce the fieldo 1Ihe po111er of this Pet hod is that it can treqt termirFit io'1s anr'l

1 ~s cont inuities. for finite ~tructures a self consis+ent approach

W\1jbe used..Assurrie ? surfa.ce '."IBV • iR e-xcited (along ¥Tith radiation)

~~ each corne r Gf the structure wi~h unknown excitation c~efficient.

For \'\., corbers ow: ~an set f'- equation t c P~lve for -'-he excita-t;ion

cuefficients (this is also spplicable for multimocte systems ) •

The problem can ~lso be solve1 in abRance nf incident fields which

i e useful in laser work.

The effect of tr._ircatin[ che guiriing sin face conductor. .no.y be

.._ calculat e-:J by evaluati11t, the difference ~ .. :tween the radiation field

due to the 1l!edge vrith a semi-infinite face , :it1d that due to th..., wedge

cb~~lementary sheFt (assuming that current ~ensities in both cases

are equal) .Reflection And transmission coeff icients have been

1 t .;i • " ' th , eva ua P~ ~sine me oao /\

T 0-.k:.1~

l't1c:dv.,r t ~ Dl. R 0

i-Ve<,'.2 ~ ~ 5 ·."" ('<\. t -~o ~ ( ~"- ~""i' ~ L~~ s \•·) c \\ L",2.\r' ~."'-' ... \\2'., e (!'.

~ ~ I C'\ \ '~ - -;>\

~ \ l.\ \' (.' ~ .--') \

o ... ~1 v et '" J ,c;),~ ~v'~,c.<L v..;t.'.'-•"- ·,,,...<-,d.zV\.1 ... . , ci.. ....._,.;;_dg-~. 1=·s· 2.4

In effect euch ~ wedge acts analogously to an Rntenna.

The ·:\ircboff method used for this calvulation ( flssuming field

distribution of guided wave is an approximation to the distribution

over an imaginary aperture) for the TM wave agrees well with the experimental resul r.s 9.ncJ exact solution (obtained by methoc:i

of images and simple Grenn's functions). This is not true for

the TE wave, where the exact solution predicts highly asymmetrical

sidelobe patterns,actually observed in experi~ents.

Both,theory and experiment preiict little change in ~~e pattern ~ith variation of wedge angle unless it afproaches 1 20n .

:n short,wedges are ef~icient surface wave aerial9.nor surface wave transduc~rs far from the edge of the ~ .. E'd~·e, but close to the guiding

surface,the am?litu~es of the excitec radia~ion ~nd the surf~ce wave are approximately the eame as thos e associated with the sane source above an infinite reactive p l ane.

.::

LL,u .. ~L i ....... 177l/ ::::: __, ICL! ~

) /,. R G.<l . o..' c- ,, (;. 1c1 '- ( le , < l-. ,_

Fo-

A.-,,-, 1)li t· dl'"~

o~ ~"' • '"''-~ ~ ... C4.oJE:.!i ....... J.1:::: /I

I .. ' ~ ~

The surface wave c~uses rsdiation into co~plete external space

3nd a reflectPd surface wavto~he ~adiation field of t~e transducer

can be neglect eri e.s an excitation of the edge since its pa.t tern

function must vanish parallel to the i mpedance ~la~c.Thie ) linciple

has been usPd directly in aircr~ft communication , where a f l ared,

rotating Purf8ce wave aoerture type launcher is positionFd <?.~

on~ Clat region of the undersile of the aircrqft

Y~h:I. ... ) ,u1 .... '"" \..1£'1 • ..._

'""""°""e,,..

A .-vv-o.~ l· <..011"'"" V•"'-o. I··~"'' b::J ci, ~ c;._,t,..~~!)._ ~\.l r ~~.C ~- INV'v• c -

I'S· :<.6 !be rotation produces an omnidi~nc~ional radiation p3ttern. Bowever the e,uide length (aircraft length in front o~ lau11cher )

is short and varies in different direc tions .Chis ~auses an angle of tilt bPtween 5 '-' - \ 1 ·s~ which int rod uc es gai11 r ipplc at the

recepti~n polnt . r his can be overc0re by cnvelcpin~ the leunchPr

b~ a dielectric shaped so es to co~peGsa L e ~or t ~ 0 lsngt h

C. ifference~ lreiucing t.ht> tl i:rec t .. ion8lity o: t!Je ~'~ve ) •

.f- '-4"'.5'"' c-~ 1 '.I\· c-:., (VI,'~ ::> ,l.:-lcbl..

c-..~C-A~\-,.)1o-~ lo.v""~e\·-

.,..c fr;.. I-; "' J c" p'. vo \·

ltt\,~~\

:3 ''~

(~·) A "'._")\'2.- a.( \-~ 1 \-. l'oJ 0vVV-.f0V\ "'O.h,.,.

!=1c.. 2 7,

Such aer ie.ls c:.re mech3.aicc:::.ll:y siin?le , flusb rr.ountable c.nd offer 2

wid9 h2nd 2nd a moderate gain 9

-~ cco1'd ing to tt c 9rinc iples o: :·urfR.c e ws.v0 1 j_ffrA.c t ion, v:e ser-:

thet the ter~~nation 0¥ 2 surface wave line iR equiva le~t to

s pee ify in j 2 -~ ie l d dist ri but ion ovsr an effect. i VF l y infinite

ap~rtu.re A . .:-,.-- . electroma.gnr:tic hor·.L1 - finitE c:pe:eturE A'

F is .<..~

1 I

I A ------;>.

If thG gu ide used is a dielectric,and thF Mode ~ e ipole,we have

a pcly~cd aerial. T~P launc~er ~ is eq~iva le nt to the t ransition

bEtween the w~vegui3e and th2 torn.The horn excites evanescQnt

radiat ing waves which in effect c~use storage o~ energv in its

vicinity,whilst t~e surface wave lHuncher crea tes unranted

radiati~~ D~eldso ~oth cf thes~ eff~cts re duce e~ficl2ncy,but

can be minimizEa by propPr impedancP matching .

'!he effective a.rea of the s 1;rface wave launc11er is ·~ f..)::i-· 2 d i men­

sional case (.~.,_ ~-- 3 d..) . oL is ti.16 tindlng coefficient

The widtt of t~P raiiation p~ttern is invereely nroportional

to beosm s i zP ;.~ .pro9 o:; ·~ i·:ir· a1 to oi... .rnbe trick t hen ie to c;fficient l y

e:xcit<:: r1 l oosely 1Jound C1 U .l'."TqC':.' N3".'e (srnal l oC)ensuririg rnin in1um

interference fro~ oh2tacles. n1so ~ ~uet be long Anoug~ to minimize

the ~ffect of evanescent wav~F on launch~r.~ut short enough

to ~inimize e ideloh e 0 ~y Jestructive i nterferenc e.

~he polyrod aerial is an example of an end f i re ~ 0rial.

AnoU'er t.y9e is the 1~1.g i aerial j which consists of "3. large 11umher

of directive clements ~ i~ effect forming an artificial di e lec t ric

guide.Corrugatc1 aerials have ale o ~ : l_ -~

been ueed,And these have the properties of 8canning succ es~fully

over angles llf to 30° from the normal to +he corrugations ,

by altering t~e phase 1istribution of a linear source.

Arl these ae-ia l s , becaus r of Atric~ launching requirements qre

unable t o produce beam widths l ess than 15°. V8r ious rnod.ifications of rrultiel ·::;ne11t aerials he.ve been investigatP ::J

the r iust succesful 'being bimon 9 e "C i :;ar" aerial '1'1d Mueller' s

polyrod with periodically s pace! die lectric discs acting as

radiating e lements . Al.su cuzved surface aerials and dual

polariza tion beacons have bePn us e d to nroduc e YJ.arrow beam

radiation.

§ 2.09 SLOW WAVE GUIDES

Of particular interes are iP1pedance moc1lla.ted ., i1suRlly sinus--idally)

guides which produce ''slow'' leaky waves. ThPc:ie ca.!1 be scanned through

any angle from backward end fire to ~orward end firef2+

1

t--,{_ -

lr-r-r--r-:-.....-...-.~-.-Q I " 1 J 1 I ' I

'::,:,,\J<.-=·.:1~113 v ...... ociv \t-\<.J. er .SC. -

c., c. ol'e\eL.v..= '- , '.,.1 - · .1 r 1 I 1S -

ror low periods of structures,and low frequPnciea,~he ~aves produced

are 382 ent ially s·.ir:~c ce waves. :~owev er, for higher frequencies, radiat in~

co~plex leaky waves are exited . The periodicity condition requires

the propagation constant fGr the ~t. mod~ to satisfy (l (\ :2. ii ! ~ "' :::: \ ·~ - .:)..., h,, (_ 2 ~~ )

cL= P'2-"'0d 0-\ ,\, . .;c,\-'-' 1·.:.. • "'1". c"'.:j ~v1h::~e.-

ln general ~~ is complex and sh8ul~ be writt~n as~ J~ where ~ie

at t ennat ion c oef~i ~i ent, ·:1'1 ich must be equal for all modes present Q

'flhe binding coefficipn+ or tri=rnsverse wave number- rr.ust satisfy

\< =J 1<.2. ( >t- 2;;- . . ) :2.. ~.-<l <' ~ - ~<.> -l I\. - j e,/ . c

This complex coe~ficient implies leakage of power away from the

gu ideoThe two equation::: may be combined graphically to f orm

a Brill-.,uin diagrarr, '-- ~ .. :;. r::,~ (::. J

.... \~ _,, ....

S\.i..;.J.,__.J v-e..:J''"°~' Cov&v-> _, ~· •f1.< t...?..­

\.--..1c, \/'-' N C Q '2.;

L) ·~h~d ~cl -I '2t k ,.j ., ... -:: ~ v t. """' d .; !>

cl

I:-iis method is a very ~0·1.erf'l, techniquf'? for determining the

properties of thP waves. ( ...,c.shi:: d lines represent "back'.vard '' spec c

h.::irmonic s ) . :nt srsect ions of "backward'' and "f or.-1erd" hsrn onlcs

&re called morte coupling p~ints.

1n the Purface wave region these create standing surface wqves

whicr: hMVS been observeC.. ~n the comrle:x v;ave region iut erprt>te. t i on

is not so clea~,bQt on~ can use the angl e of radiation 98 a criterio~

The latter i8 de+errr.inPr1 by(..._.':- er.\. .:: 1\::_rc,,, •• 1 e11C' 0 points ® 9.nc@ currespcnc to baclc-:~ra and f'cwwgrd endfire res;_JPctively whilst' CD represer.::s b1o'ldsic.e radiation.1· t eyac+.l,y the 'broadside point

we fin:. ol-7 o : ·1· t nc rs.r'liatiori occurs.:'!:'€ 7'rillouL1

CiagramLiatic methods h3v 0 ~e~n ~~fi~ed conal-erably 5 and quan~it3tive

results bav~ b•en a~rglnei.

Cnce again care ri.uet be ts. 1<:'"'n to inve2tigat2 ~hi:> corre 1~t T iem"=J.r_n

surf3c~~,3nd at ¥i "Et se1tral ~R1 ly ~io3ec~s o~ this work o~tPi'1c~

s:mrjcu'2 results,·::.....,i~~ ·::e1·'2 ·1·t2eql~ EnI;ly ~·ejEcted.

Co~ai~er t·vo dielectric sheets.There qre t~o modes for each pol'uization - sy;rr.i::: +ric .:n:.d '3.ntiCJ2rrrrietric , eac'.1 1:.ith a differe·1-I;

1·n:1v e 1 Png th.

/ .

/

1=" ~\

25

'he effPct of coupling c~n be estimAted by thP folJowin~ qpproYimete

mgthod . Consirter s surface wave rPeonator close to b ~erfp~+l~

conducting )lanP.

c; l "=' ''-V'I'; >"0

\ r• ~,..,,e l c,\ I St' le.< .. t> WIQ '\ 1- o4 CA>vicL ... u-: "'.) ylc..1•c:

~ Ccr,ch.1cl JI<) plt,;·V><c. ~ u- i)\,

Tr a 10 2'" lPss F1 8 . 6..L

rs-~r~tor, ~ction is inv8ri~nt unier slow deformation

~ ::. ~w ~ , w

!er ? 8onst-a'1+ rhis wo~ld corres~cnd to thP antiFy~me+ric xoie lzeTo R~plitude at distan.::E .t). "he change in energy o+- ~r,e P,ystem ~e th~ 1~ 1 ::ill is b1 ought from .t..-cSt. -l-v t cen be found b~r intc;r!1t ing -l;r"" ralliat ion

prP3SUTf' p::. ~ Jvic H-t<.. - j__ (: .... t.,_ ~ c..ve:::-' plqtp PYeA Bnd )VE"r

distance frcrn .t.-+~t ~J:... (1 s a•1 2.'."'proxir:mtion ·:,ssui""C tl1qt E..,, Ho::

o.re h'llf of tne i r unnerturbed vr1lues) •

PRlng ~},E sam<: epprc..,gr:h .v0 C'l"'n cCJlculgte tbe vrnr1<: don"' l:y moving

the resonator end plqtes through SL • By cl fT i table choiCP of ~ L

r:e can ar-r8.ngP thPse t·vc ener3iPS to canc"l p-ivin~ S ""-'r01 " 1 -= O

'This ' 1 ields ~l.. = ~f\ :i:1d ~ience "A cRn b"' calc1Jlc.ted. (Th<> eame t~1 I1e of ,1 '- )\

cal culation applic~ to thP 8y-~etric mode , wi t h e magnetic wa ll

replacing ~he conducting p l ane) . result~ ch+8in~c agrPe well with

exact theory . Vc. . .-'.0<.\-,c,, ot vv-ode ~"o.."e..\~"':i"h

vvi 1-\-i !>e..~0..-0..b -~1 ·· o\ cA., e..\ ech-, (.5.

0 c,

0 ·8

C- 7 !:.__ _ __. ___ ._ __ .....__ __ _,_ ____ => '> ...' pc. n::;'" \ ~ w 1 \ .. _ , ... ' ( Id .. ) 4

r- '<j '?,3

Simil21 ana lysis can b~ ap~licd to dielect~ic ~o~s. For cert a in cri ti ca 1 guide d if~;(iens ion3 ,_re phase d if f ere nee bet71een

the qymmetr ic 3~d antisy~metric modes ~ill be 2~~ 8nd beats ~ill -.. . = As A. o... occur . J\ °'

0'2<.\\' )\ l - )\, 5.

1-.i> r os"'t wavelengt:1 i.:: irrpo.:·t::..r:t in desir-ri c.,f coup l ed - guic,2 directio:1.<>l

couple~E.

~s an ext ension of the p~Avicus analysis consider an infin ite

hexagone.l array of indenti.c".ll circulR.r c. i electric fibres (equally

excited) - a crucie model of reprPsenting light excitation of human t ~·)

ret ins::i. l f oeve l c onee 111rhich ere "iI a.nge1J b cxagonally ( approxirne t e ly) •

Son9id2r the sy~metric mode. The d ispersion relati >~s ren be obteined by evaluat ing the rie l~R at the surf8ce of the central

fibre as a superposition of the individual fie l ds of the other

s i :r- (by using a tensor fc£m of Taylor's exp8nRion ) . 13y in~isting

that the t angent i e l "'ie lds be cont im10u s across thf d ielect:r ic houndary , we obtain a Bet of rRther a~kward trRnscendental dispersion

relations. J:ovvever for c l osely Rpaced fibres t hese ri::<iuce to the

same form as that for sin~lc fibre transmission.4n~ly sis of the

antisymme t ric mode is even more corepl icat ed by the non-cance lla t ion

of h i gher order terms,hut for this highly eymmetric case i t is

thought that t he results ~i·c the same

OG

0 (]

~.103 tr· ~ IC TICN 0 VIS!C~. IRCCF2SES

As mentione d before, for research into human vieion,we should

conRider conical or tapered guides rRthe r than cylin j ers.

Analysis of the open tapere1 guide can be carried out by a

limit ing ~rocess

·me. ro.pe,- ,s c.

l~ V"'-·\v. "':::> :::,·, \ vo..r. 0\\

a. s 4:>.-0.. J 6. :, ·....:>/ 0

D i Q..{ e. ck ,,. ·, c.. h:; ~ c V" o. e f .,..._ x'' v"V-L'\ \" e.d la .:i C:·\...

>;.<.Ir o~ hue.;,coe~'-- (.;:i\·,.,,c{12.,,...,

:onsidering forward propagating mod es only,and neglec t ing the radia­t i.on field ( as ment j_onl':ld on 9. radiation is negligible for res sona­

ble tapers). fa fter applying field orthogonality conditions 8nd taking limit s an equation 0¥ t~e coupled transmiesion line form (matrix) is obtained.~his is a surprising result,for such equations

arc readily soluble by standard elFctronic transmission techniques.

2. 1 04 SHI.CL r13:D '.r\ VES

To conclude this discussion o~ interacting s urfe ce wave guides,

we shall consider shielded waves ~

The 3implest case is a wave excited between two infinite paral lel

plates. '.)(..,.

·l'

}\.__....;;;,,") ~}

F1~- 30:

Impedance boundary con~itions are

'=..: -;;: z_ "·" .j--::::o c.1,c.l ~ = z ,... v- .'.:1'- c.~ I~ ~ ~ ~ · · 1 ll "b ~

For a bounded structure we may exp~es2 H\

o: liscrete surface wave modes

H i:i =' ~ a n { vv ( d ) '2.'l< 1::> ( - j }\." X. )

- t-1 o..;< w .Q... 11 1 ~ <.:: l\, 1.1 c . 1 ; c ' , , , e ( /\ ,~ . · ·11'\. ( J J -::: vf... <U)( ~ (\A 1'- '-J ) t-

f'v o \IV\ E = -. ,_ cl-lb-" ,._ <l"'"" oj

i::-"" I - 2;. o . .,.Ll.A....,,/<l~E ) ,.. (co:..- p) .;i_,,._p c- r~"" ~(_) :i ?> ~ :. o ~ D ,, ( v<, -r (!, ) ~ ~ v~ ( -j /... ,., x. )

i.A,.,,(<>l.+ ('=>) = Lo SwE: o<.- r-

a.s a S UTJ

k:;.

A \:so

~ a "' ( Vl "- I A' w t_-J [ ~ e--,,: ~ (- tA_ "(._,!__ ) - r Qx f' ( V\ ~ d_ ) J ~ C\v'l ( o( 0r (- v..,d-_) + r- Q..-'A.\' "-"'-"'cl) J e~p ( -j i'\"' Jl)

.A-.._ [_ <><!., 12..x(' (- .2 ... l._J..) - ~]

""- '2-x p ( - 2 IA. "d. ) T" rs

d._ = .i_ j v.... ..,. - ~<-L7. <-

""-"' 1- j t/J~ :Z..,.

~ r~~ ~~

(\ "' .::: r( <.! I -\- f . ._ C\ r. c.l \< ~- \ "T"C'v '-

j.;\.,-, s s;·,5'''-~..':1'V\) stuw"wo..,.,;~s.

c ... "'d ..

• . ~-!;> 0

~ ,,..,( o) "-4-r(- 1<.c-v(d - -a))

\ n"f\~·IY'\:J C,r.c;,v-...p\dG.\_j V)(.)H-;\f'.~~<_,,_. .. ~(._1:- · v;__j hJO.V<!~

Fer o.v1 c-...C(.v;ro,\e, c_c.., \vvlo.\, 0 ,., C;:, 1 1 ,,.,olo•' c, pe1l·L_...bc, l1 C1-. tc. c.-h1 .. , ·1Vil-(C.

\"'-

LL .::... \<. f:> ·t- $.; (.. \<..p ) v-1 i \ 'l,

(S,l"l,:\c-.,\~:l {o.,- \t\e_ o\1'\ei-

i , .. , \ ~-o.ci: 0 I'\

<;_ L u..) = v<.. +- kp ( l.t 1"" k<} ) 1...\. - 1'-Cf;"

KHS ( 1 ..... :;(.._) ) bj l<p

~ ( ~ t- ) ~ ~ K p ( ri--rq,. ) -Q.)<.. p (- 2 k1:.c-() ~> - CV

[ G \ Ji 1 Qv1 .) , b ; ~ Q l'\'l~_c::,).v i'l2 .. 0~

L-),. ( \/~) == (u... ~ k \" ) .e..~p (- I.A. cl ) . -~ >..: s \;.,"' t- G ~c.I er £ (.) \ \) \ . 0 ...... \ ;,.,

U- ~ \<.p ± 6 ( Kp) = \< p (I +- 2 e.h, p (- l'-rd) ) , T~ \-1.,.::.. (-v e) en- (+'10 .. ')

'<Co\- \S ~C\.\<.f.! .. \I'\ ·\--.:r ~ .... 1-h \"ri12.. 1r-1.·'l v<?S.. c-1~ 1-'h""'- ~we ""v •~~l<.:'.~..J o....

$.jV'A~Q....\-Y-~C V'-tude w',i\ OCCvr) C,,,,,cl :~ ,-oo t S o{ r1.·.(~ v.!v<c_v,\- c\5h<. C\1~.:<...

rC\ k e "' , Vh ~ 0\.V) hs::J""'.,,,., e..~·.,.. ·, c vvi oct e... w .\ l .,,.-c...s.1... i \- (; d <?v1 h c.cJ \. o I-· ~,"'-

( 2 3'1)

fr•· ·, d € I S \' VI -

'S . 2.( ) --- ~ ~ ..... i ~<:.C:...

~ ~ ~ ( 0) '}·.J·d <'...

, .-_ (~.__. 1- r 1. rd 1 .

§ 2o 11 HARrlONIC .HTAL'{SI:3 AND P ULSE F.RCP.AGA~CION BY St.ffiFi1C E WA VES

So far we have considered h8rmonic surfsce wave propapation . f!. n impOJ ·tant aspect especia llj· for the

study of transients,is the investigation oL C.:2.5

by surfac e wave modes. As usual i n problems pulse propagat i on

with non-harmonic excitation i t is customery to apply invers e I'ourier or Laplace transforms to the solution for harmonic time depemdance .

For " open " structures the steady state response is given as an integral in the compl ex number plane 9

Surface waves occur as residues of poles in the i ntegrand$

For a slit - excited grounded slab configuration TM surface wave we have

<J' J) ( ) nl <, [ I )._ J... J ' 1- [ ~/ I.. \ .. ' ( -'-- ,-.. ) 1,_ <... x ') s L -:= V"'"-C_ l 4- i;;--?'f - n." {'-:> -f<.."" s,,..__., :.1...., • ;" - !"-..., _) «--x-r d _K<-,c'--- ~-:i~ .)

i,,.;\\-i-, k. 0 :::.c.,..,/c . •fh "' C\yji•<JP•;r,h:. d;Sf<",-::,;c.-. ,-., ,~,I I "' I s

.:t_ .J f>:i.-k. 0 '- = f E k . .:..2. - ("' ,_ ·tc:-.""-(c:t J~ k., L-_ f:''-)

.x i ,!,

L )._:... < I l I<-: :S ,- - (3 'c,

rf : ><- 0 (c>.l'e -\..,, "-)

.ho

1'@-~~~--~~~~--:----:-7~ ·. .(), t. \LC r,.;c.

6"' \::..~<." ...... ~ ......

l on c-h,c;. l·: .- :_) t~I "•" i

1--o

a suitable transformation which

dispey.-sion re lat ion ~"° h,,J!Z

~ I _/ / 1 will d isentagle the transcendental

R.w 1 ·, "' e..,, c.<ve ­

t;, ·,1\0..,·,,,_ 3""'"'1'(,.. ~ fS \1. •. }~ ~' n~;;.. o_ re.-

-~O

//J J / /

C\~::)'"'f1Vu \-C.~

f..... k. C...<€. C...v~I h Ccc ( C. l ..) L"L

//~ V41 J( .!> ( w r c..:> -~-\ '> ) ~c' Ve. v \ '--'"' f., v"'- ocle .!>

)<.._ , K_ ._-------~ 1;. ..,

The Brillouin Graph For Dispe ~sion Relation t= , ~ '3S .

Use the following transformation

I. - l ( 1-o.v--tt=ol..\ 'i-+-o(~

'-o - j ~ ) \I I {A,>J e;, -1

6 - I ( \· c. V\ - t E-~ ) J I -t- (;- ()\,, L

\- - cl JE'=I For complex of., choose Riemann plane for which the square roots

are positive definite on the Re(~) axis. Therl~:f~ a I~' m~pping and the integral is evaluated over RE (~} axis. E ).. ( 2> ,+) = @~ L ~1i ) ~ (""--) .0xf L} E L;;l 1 ~ , t) J cAol ] (:2.A-4-) A ~ymp~otic expressions for t~~s integral. may be obtained by saddle t-01,.,r , .,..\"c~,-"' 1;-,c,.,.. ( \"\1t::.. cv-1~ ~' ? '"'-'J- , <:CA 1,.,,\· C..ch\.-•bv 1 ·,c,.,,~ o.v-~ fv-ovv\ .h0-dcJ..1._ points situated on the reai axis. ) For ;.t =- 1/c one pair of first order saddle points -r oc01 e.i::::it ers the

real axis at the origin,and with time moves away. i-.A t ·t -=- ~'i<S/c a second pair of first roder saddle points ~ clt:l._ appears

from +OC> . At a certain time t~ these two pairs meet 9 and one pair of second order saddle points z~y appears. At that juncture the saddle points leave the real axis a nd t r marks the end of the asym-

totic transient response. For higher order surface wave modes we can still ap~ly the same

technique, but now we have l<c~= a.~ and hence require a more

general transformation

kc = J. 0 J (co. ..... -1 >(,j._);,,, +- "'l.. iT i.. J ' +- "'- '-

j i + ( ::"-

Whilst the foregoing analysis appears to have little physical significance,it is most useful in interpreting interesting

physical phenomena involving transients in dispersive media (_ such as magnetoplasma) studied in a latter part of the paper.

§ 2.12 APPL I CATIONS OF SURFACF. WAVES TO SCATTERING PROBLEMS

So far we have considered systems where the sur facP waves were guided

on structures with an open topology.

This situation is quite different in diffraction of light by cy l inders ('l.7)

and spheres.Consider the perfectly conducting cylinder

(. ..-..., ~s s.~"!C I ;o ,., o~ c ,5 i' '" d <?. 1- $ c_c., \·re ,r\v-..'.j \{-, Q ·, 1~ c,, ol. e1" \- vJ o. v-(:..

; ,::i sci

The... O..Vv1\'.)l\r.;de c.-..v Q VY'lt- 1 l:>c.. ~,va. lvo..h::d o.s

IA. ( Q ) = 2 vi c ( Q ) -r .L I ';) G ( 1)1 Q ) d.. ,JJ . ..21rj t;j.,.. ~?

;= '- "' p

::,. G i-e..~1ri is -~_,,,..,c..I ;o.,. _; G- ( P, Q ) .,

A\ So we_ ol ~ ~\vi.z 'V = -~1_.~ \JJ Q, ·= 2.. IJ. 0 ( Q > - ....L_ s _ J: ( ,,, Q \ °' { .,, c ;;£ 4- 7 ) v ) ::i...11 1- ..:)\'\... '

After laborious calculations and asymptotic expansions,Depperman (:i.9} /.. '/::,

and Franz arr i ved at the following formula valid for -~ · ... ·:<.t<.. <. 1.._.24)

v{Q)""'- I )?3E'.1,<p[-j.x:.tf-2·3·s.<...(i:_.G ·-.--t~ ) ~v-..'f (~ 4 f'1?>] (:~.4-8) (A simi l a r formula holds for the sphere.) ( Thus even when Q is withi n the geometrical shadow, a kind of

"creeping" wave wi ll reach it ) . This wave may be thought of as travelling around the circle of radius

R wit h v-.,..,C<:.<L = C-/(1 + o 40-:5 _,Z"~3A or as one travelling with i.:r1> .... c.i..::- = C

but a t a dis t ance K.(1 ...... o.40"3.)1...- 21~) from the centre of curvature.

It can be shown that this wave may exhibit both polarizations ( t: 11 t. tc

s" <fo._c1Z...) c:t"'J H 11 ) but the latter is by far the more dominant. In fa. ct, as R .-::;;. o0 ; The \-1

11 wave degenerates into the Zenneck

wave ( as can be seen from the formula ) • We also observe that the wave is a t tenuated though no joule losses occur ( perfect

conductor ) .This is due to a process called " spraying ", which

can be imagined as a cont i nuous loss pf energy at a tangent

to the direction of propagationo Another thing to note is that for f large enough the form of

the wave becomes independent of the method on injectiono

In fact ,, we observe that a wave travelling through 'f'= iT

is attenuated by e..:xp[-2-1C/>--Y.s]so that it will be important for .x.<4

but negligible for ::>.:. > Z •

It is this particular wave which was first observed experimentally

( indirect ly) . The reversed wave ((f : Ti ) must be considered when

calculating radar back - scattering cross sections from cylinders. :.?.. .., rT C-rr) + T (11) l

G- = L .-e~lc..c..Ve,:..t S.vrf(;;..<.<l,

I TI"~~ le...:.t-=\ I 2-

T \..... r_ r~\ v:.Ne..d > ""' '"h-~(. c:..,

C.v"e, ,-e_l?\ecied. O.V\cl Sv 1-~>-C e.

!;VO.. v e 0-. ""'f I ; h.,1d <:. ~ Y' €.<;.1')e.C{-°. v QJ ..::\ ,

Where !.'" is ratio between real cross section and that given by geometrical reflection.

TY-ef i (Tl ) ~ ( °+A) y.,_ 12.x p [ - J o/4 "1"" ..<.j >'-] ( 2 t:; o)

T I - ) y.!.- Y;, [ V5 · f - - + l ..., 7 '/s) ) .Su v·f'Ct(.Z (ii) ~ ~ . Ct.2,1,._ '/.f. :X.. e..xp - ::Z·2.Cx,_ - ~ L 1Y~ + 11 X.. '-<- )L

show"" bC. \ o 1,J

~ ~

6·

!=I c 4 0 :J

LL-~~~~-=--;-~~---:;:,-,--~--:-:~~~-:-~?.::X:.· The experimen a results confirm this behaviour, (even the position

of extrema in the dominant H" mode) for cylinders,but one or two points of disagreement occur for the sphere.

The case of the perfect conductor is the only one amenable to such an exact solution.

Scattering from dielectric and imperfectly conducting objects would be complicated by joule losses and curious phase delays.

Nevertheless,the surface wave idea can be used to explain two importa~r

-11 paradoxes " - the existence of the glory for water drops~- ( .,... = 1 · ~ s)

and ripples is the extinction curve of the sphereo

The latter is complicated,so fo~ brevity consider the first oneo

It can be shown that for fl = 1 33 the simple glory ray see (fig. 4 l (.o.))

cannot exist.However the glory is observed in certain misty

conditions in the mountains.This could be accounted for by the ray skipping ~ ~ round the circular arc t as a surface wave ) ----such a phenomenon has been shown to be possible ( i= .0 .+1 (bi )

Tn.1.z.. 3\0•-_~ ,.,;: .'j ....... •"Ovs" C.:. d; ek.z:d'\,.: C

Sj'ha...-e. ,

~--

le.n \·n. ~;v c_ Q..:>c plo.'' c,_\; 011

0{ \'he_ :J k.:.Y:J \'C·_j

\.!\,.,« ~ h ,,,,C\ \-e,.- d ,." f ~ lC.l ) t=\ 5 ·•+ I (b)

The above argument ( as may others ) is entirely speculative,and the field is open wide to other interpretations.

§ 2 .1 J APPLICATIONS OF SURFACE WAVE TRANSMISSION LINES

Surface wave transmi ssion has been attempted between ships at sea,

using water as the guiding surfacP for Zenneck waves. ( This fai l s

for very stormy conditions) .It has also been ased s uccesful ly in communication between moving trains i n J::ipan.

In that system a Sommerfeld Goubau wave is transmi tted by an inverted c orrugat ed Y guide as shown in fig .+2

- w .. v-r:·o:.c-,l...-ro. JV. ,_. I ;_,;_,\'YI.""".)"' ~(.;;-'\-•t '-' >\;,

+----+--- w""h--o l

(b) Tv-C.IV\

Co'"'"''~" "·,u,_\' 'o '~ 5...,£-\ e '"" _,,.

Losses are reduced c onsiderably power passing through

the guide,automatically travels through the trAin a sort of self

compensating obstacle .However spuri ous obst a cles may affect

D "cd G:::>anc,. \ SJ r{c.c e rJo.v~~ ,/tclc;:.

Cl Mo...':J'"~\-,c. ,,....e-kl - i:.\<!.< { tt L ~.~ le

The single transmission,

but also as a link between a transmitter and an adjacent mast,

or as a coupl i ng device between a field strength probP on a baloon

and a receiver on the ground ( at the same time holding the baloon

in place) .The drRwback in practical systems is the bandwidth

30M1·\~ -3000N1-15(Spread of field excessive below SONH.)attenuation

too high above 3000H1~1) ,and the exposure to weather conditions

(which can significantly alter the impedance), bends and obstacles

on the line. The line must also be insulated .The optical fibre

guide(d i scussed previously) suffers from perturbations due to

supporting dev i ces. However this can be overcome by coating a thin • • • ( 1'1 ) )

guiding film with a supporting structure. (The optica l microguide •

The dominAnt wave is the planar~nmode which offers low attenuation

for thin films,tight binding for thick films.

This system enables the wave to negotiate bends virtually without

loss.( attenuation of10do/ K~ can be achieved, but is limited by thermal and quantum noise ) •

Under certain conditions the presence of surface waves is a menace,

and means are employed to eliminate themoThis is the case in certain phased antenna arrays ( Yc, ').1 - Udo.) where surface wave resonances decrease their efficiency yi..~)

This has prompted the study of surface wave resonances on phased arrays _of rectangular waveguides with and without dielectric

.ll} plugs ( experiments have been carried out J •

SECTION III

§ J. 1 SURFACE WAVES ON FREE PLASMAS

J.11 ASSUMPTIONS

In this section surface wave propagation in certain plas~as under specific conditions is studied.In general solutions to a plasma

problem can be obtained by the method of orbits,or from Boltzmann's

(or Vlasov) equation.Under most conditions both yield indentical

results, but the latter is chosen here for simplicity.

The plas mas considered are:

1 •

2.

J.

4.

5.

6.

7.

8.

neutral over dimensions much greater than the Debye screening distance

temperate - i.e. (drift velocity of charged particles<< .r.;;:r c.J thermal velocity ~< phase velocity of the wave - 'J <<..J¥<< - J

!> '"'~~ W'I K This is required to lineArize the Boltzmann equation and is satisfied except at or near resonanc e,where c;_ ___,,. o

Effectively collisionless i.e. f r equency of e.rn.wave >> collision frequency.This is eertainly satisfied by submillimetre waves The negative change carriers are electrons,thus implying and wi_ << m + ) '-'-::.,. ...... .,.."' 5;.,~~ << w-'-~-ov-. This means that at the frequencies studied ion,acoustic and hydromagnetic or Alfven waves are excited with neglibidle amplitudes and hence cannot interfere with electron plasma waves. Surface tension,pressure gradients and gravitational potentials are negligible ( otherwise quite complicated relations result )~'4)

Boundary l ayers are neg l igibly thin compared with the wave­length of t he guided wave (i.e. the inhomageneity in particle density is virtually discontinuous across the boundary­essentially true for pinched plasmas.)

The wave amplitude is not high enough to produce appreciable self-interaction (discussed lat~r)

Any static applied fields are much greater than those of the wave. The abovementioned requirements may seem two stringent for a practical problem,but in fact the results agree well with experimental data.

§ 3. 12 ( 35) ~{3' )

SURFACE WAVES ON HOMOGENEOUS UNMAGNETIZED PLASMA

A Consider a self p inched column of circular cross sect ion ( v-o.c\ ivs b) in free space or in a larger waveguide.In order that a surfac e wave exist,p-the transverse wavenumber (see Aopendix I V)must be imaginary,and so must the propagation constant~. Usingl$=-jf-> in the dispersion relation (.A412 )we have P ~ °"' - r-"- + k OL. \(.II

For ~<o,the wave must be a slow one. The complete dispersion relations are

~ 'l.. ... k r z.. -p ::: ~ - ko II""' \Y15\de.. p \ c;..S,VVI C... } ( )

~ \ 0 1 .. .... ~ T - L I -°}r "'f-. - '\< 0 = Ovrs.\ o<.E.. ?b.Sw.o...

The field equations become t see Appendix IV) '\7 l.. i=: .. + ~ E ~ :::- \7, L H ~ + p 2.. H i.. =- 0 ·, Y't s. ~ o4. e.. v, 2' p 1 ' 1 ~ ~2. E),o + q,2- £1,o;::: \7T"2..H1,o + "\-' .,_Hi :=: 0

1 (10 2 )

Suitable solutions,ch osen after satis f ying re quirements a t v-~o

0 "1ol '( -") oO Cl\.V- e. -

E-i\. =AT (rv-) e. ~w. .P H;., = o, ( P v-)- A""'4' lJ - WI > 1' D -l'vt .._

E 0 = C K ( T r) e ~""' .p H 0 = D K (_ T ..- ) e ~""' .,

8 ~ / b M } ( f0 3 )

..Ll'Vl K WI o.v-e., h--l oot ifie..d. !Se..S,se,I ~vvi&:o>".:ls. Tangential fields are obta ined from(A406 ) .The equations of continuity lead t o 4 equations for the constants A,B, C, D. The existence of a solut i on imolies adeterminantal equation deduced ,in Appendix V (K " I:,'_ K .... ' )!• I ... / _ )<::.~ ) =w..._R-.../T )2.(~-~ )"

II . f7 I ..... l- r I K. r \l<o b r I.... • . . ~ "--rh ..... ...... (10+ For f i nae pendent fields m=o ·

k 11 ~ I, (nk> ) == _ l< 1(T b) ( f;° q..vC\r ;OV\ fer ~ lrll'.7\v-<.s) (105) T a( f1b ) k

0 (Tb)

0 R T I. I ( l"7 6 ) I< I (_lb ) r - -

T0

(f"1 b.) K0

(T b ) (106)

( 106) Ca vi V1 o r b e. So. ~ l sf;e.d., So V10 sv..-fa.ce H IN O..vQ.S: c..c.. ..... ex;st- .

E W Ol.Ve_>. Cc--" 0)(0

1$ r OV>.~ I{= K,l < 0 . F ...-o~ ( A 21 t) we.. s~e. l'ho._ ~- ~is

OCcu...-s fOv- w < Up . A v-e.50V"'o.6ce oc.c u..-.s; f 0 .,- K :o - \ ; -fo.- 11\.i e..""'

r I T -:> oO ( v- t'. So"' C\ 11 c e. 'i > ~ w :::::: cJp /Ji_ ) . L o 1r.; \ r e..cy.i en c....J cl e., p e"' d <2-- Y1c.e. . J_

( !'V o y Q. · . - .f o.,... x_ < < \ J K 0 ( ")l..) -'"> k ( )~ ) \<. J )(. ) -=? :>L )

A \- \ow fv-Q..C'rve._V\._,~e. ~ t.,, "'-' - ~~2. _) r I~ \ o.r5e CAhd. T ~vv--"ll - we,_ 1 l, ( ;-r b) _ I

w ..._ r ""' T0 ( 11 b) (Tb)~~ lTb )

S, .._ih >\-: )·v h"'~ f ov- I', T f v-o""' ( lo I)

- ( ~ '"L - Ko "2. ) b "- -t -0 ~ ( J ~ "'- - ~ 0 i. b ) = c.J z b

11,~s ·,s 5°1vvip\;.{:.~ed Co\/\Slde....o.bS

e o . w P = f :;;ur "' to ill- 4'.:iv- Q..

{N ah: ... ·.- fo..- x.-;:. >t _, I 0 (x.) - ">

: . - (0' "2.. - 1~ <:2. ) b 2- {o 6 ( J 16 '4 - 1<: b) = ~p . ~ 2

,__ fo f>

Fov- ~ ol~ 10\re.. p\CA..~\/V\.<A. ) o..""'ol SVv\.C"-.tl b ,

( N oh: .. ·.- {o...- :>C. <-' l I0

( >d ~ f ) ..L 1

( )d -~ ~ )

.· .-(r1:,'- - \<\) .,_ ) b"l. ,{,o~ C '1r-1.-k0

1.. b) : 4-w/wf.._

I f 'vh e f>loS l/V\ OI. 1s; suv-v-ouV)o4ed b-21 C\. C.._j,;V>°'v-;c.o._I _su\ok ..

v~e., Ei =: c [ r M (Tv-) K""' (To..) - \<'.""' (T..-) I""' l To..)] = &""'""' (wl,,;~ v~\l\is'-" ~'!"o <AY o...- ~s Y~v; v-e..d. . )

)

The same kind of analysis yields resonance and cnto~f conditions and low frequen cy equations. Thus we obtain the ~versus w graph. A similar calculation may be carried out for the plasma embedded in a thin die 1 e ct r i c of p e rm it t iv it y E • Th i s shows th e poss i b i 1 it y of backward waves as shown belo ~ r

I

(Gt) - f"..-ee. c.'1\i"'cA r~c..o.. I p-\°'5 >""C...

{_\:.) - ~\t<S,l'\'\A eM be..dole.ci ~ h

0.. °' ; eJ e.c...hri c . ( W\ ~ 0 ')

-Da.>l.e.d ""'v Cl.re.. low O-.\f\~ hi~\.-\ f ..-e.qve."' ':::1 ~>~w-p\-oh .. ~

( b<;.c.\::.wc...-d.. WCA"E... v~\OVl

w::;;;__ __ __.__.___._ ___ w _ (;jr (lo) GVP/,"; ~ i s sh o-d. eel) (a) °"'r/.fi_ n~ 45. ~'.'"le '-ri.

A better example of backward wave .. s is urovided by the parallel pla"'e plasma slab.

f"" ...

. .. . · .

Fq;).4-6 Once again H wave solutions cannot satisfy the boundary However, two types of E waves are possible(see'fig 4h) It is found that only the a n tisymmetric mode can display wave characteristics. ~ T£e solution to v_; E~-i' ~=0_,is Ez=Asinhrx (Antisymmetric Ez.=Asinhrd ... expf-T (x-d J) - 1~\<d..}

.l. { 111) E =A s inh rx. - -:x... -::> d.. z ne. bovVlJ.CAv.J COV\ d ·, 1';0 W'\ $

!! · ( Eo ~ 0

- Eo K11 §.;. ) = 0 (112)

Bv\r f..-oWl ( ) Wf.... \..,Cl v~

f" ~ :::::. _'I V E ~ o."' o\ £To = :!.... \7 T ~ ?o -T r.... T i T"2... 6

(\13)

(S_lj~me,~,.\ c.-)

conditions

backward

about ~- o)

1< ,, Z> A ~ .1 ·- \ r ] ~ 'd ( - - ;:> ""' ><... -· ~z.. - A si"-t-. r.:A r'l... d')L )l."'c.1. 0)<..

~p(-T( x-ci))] >l.::.cA

r tc.."'~ f'ol =- -T ~ n

· ·. So Iv 'r~ o~ s f'ov- \..,.) T +ve occvv- -fcv- k 11

< 0 1 e . w < c...> P

CONDITION FOR BACKWARD WAVES From app.(iii) , backward wave condit ion reduces to

-w -w < o o.hd w6 - w'(ll\ < o MT E ~ T b

c- · · - E H - O - ~WI-\'\ = 0 ~ 1 nc..e ~.s. is o"' wo.ve. > 1 - ~

We, < 0 7 t w EO\J'y5"\C'Je. l < l w~"'!>\cAe I (115)

(114)

I T T

Re..~ \\i~~ """(... cl.e.~"VllhoVI of' we,= S ~ E.0 §~ %:'.'.. §:T o(A ) we- ~ctve.. 2

) 0 i ~~ \4

1 K,J d)(_ > :<_ r10

1 ~~\~ cA,G J ~

I k " I r I 2.. ,, d s 00 I . I 2. n f r ( J -' - G<>~" :>C.. x.. :> -T'1..S'"""' '°'-Y.€.-)< PL - :2..T x-c{) o.X r'2.. 0 cl

Svbs\--:rvl-~"'.S fov- T fv-oW'I (114') we.. h01v<-

i:"\;\,~ (_rot) [ .2..taY\A-.. ricl + rol J > c.o ~ l..rc:l

This e.q,va.,\.-~OVI ote...Y-e..,.VY\·lV'l.e_.s ~e... v-'\S;OV\ ~"'- r-1

c....J Sp~c..e.... oviz..v t..v"-~C~ bC\..c...k.""-'a.vol W'o..ve.. ~0.v-CA..c..tev-i s\.r;c.~ C..V'<.. o..ll Owe.al. I~ ~12.- ·,v'\e..cyvo. t; Y.::J ~> >o\v.e.d ~t>V"' V-) w e,, <-c\"'- f i .... ci v~e.. e..,.:. h -- e..VV'Q... po '1 V\t-> w, Cl.ll\d W <..

(se.e.. nexr pa.~C2..) w"'-~c.1- etv-e.. Solv V-:oV\.S lr-o ~ =O. \"\,e._se..- p0\1AV<;

tv\o. v-k. ~ ~ \-v-0. \t\ S. ~ Y ~ 011'< -\ v-OW" f O Y- W Q. v-c\ \-o 6 O...C..~~ 0.. .,,.cl \,Jc:.. v e.. b.e...l-.a..v i OV'r

a.v..J. vice... ve.vso.. .

,,. ,,. /

,.,.

/ /

/ /

/

_J 00 / ""~ /

/ ,.,. /

/

/

-----(....), c..>p w2.. WP

.Q"

-- l<od. ->0

F':I· 4-7. ( f-> "~ C.V 5..-cq>ln fov- parcd\e..I pb~e ;.laswia.)

(Nore. I-ha\· vi ear Y"eS.oYia"'c.e. 11.,e.,. wavt.- is. Q,5QI V'I a forwo.v-ol. o""e. . ) .

37

Thus E wave has its phase velocity parallel to direction of power flow,whilst insice it is antiparallel. Hence the backward wave condition is reduced to the statement:-the absolute value of power flow (time averaged) inside the plasma ~greater than outside. § J.13 LONGITUDINALLY NJ\GNETIZED PLASMA WAVEGUIDE

1--Bo -,-~~~-!:-========~~~~~--·,.: :;:.; .. ::> · -~:· ::··.-:"_':. :'.:- .': .. :. :_ -·· -~ ': ·:-. Pl~~"""c..

Look for surface wave solutions of the form e.xp(ai.4-) e.x'p(-j 0-~) Outside the plasma we may have a superposition of E & H waves.

E wave :- Eh == £~ .. , f.:x..""' -!f Eb 0 , 1-1~ = -iC::0 E1,o (117)

where E10 is an arbitrary complex amplitude. The above equat i on ie obtained from a modification of (A 406) for free space.

H wave : - H i = H 7 1-1 A = - j ~ H ~ = A c.1 f' 0 H ( 11 g) 0 0 ° ) ~ 1, 0 J -~ o(. ~ 0

Inside the plasma we must have t= 3i = E 7,. -t- i; ~ '2.. , according to A 409 Assuming no y variation,the only solu~ion accep~able - · ~} for surface waves would be of the form E -== (E :t e,"d"•x..•E: e...-Ak.2.'c..)e.J ;\

(where K1 and K:t. are roots of A 408) 1' 0 1 S1 ( llC\)

E :x.. and E ~ fields can be obtained from A21 ~., , H can be obtained from !!- x E = (..A.) /"'o ~ where ~ = ( 1<.x. , o J ~ k ~)

•. e. l<l(,= P ' ~~= - IC~'1. -The dispersion relation ·is with the abov e replacements. Matching the tangential E, H fields at the plasma - free space boundary yields the determinantal equation. (120)

E'a (I - ~l<:.,l.. )((I,, 6'.x - k:11.., - j ~o,_ )- E'<J {..1-·Jy:_x.1 )(IS E .>L - l<:'.x1...+j~o)= o EJ,'L ot.. . I E.}1 a(. f:'l- 1 o(. Eli ..

( I A 0) 2< (_ A 4-1 I ) o( e. re r ""' ; V'I e. I< :it. I ) K "X l.. ) rs , O(_ •

A resonance will occur for ~-=>CO .Examine forward surf ace wave resonance only. For ~~oo dispersion rF lat ion becomes K~ + 0 'l ( ~: + I ) )(~ + ~: ~ + : 0

K~"l. - ~2. ov- - 1'12. ':: ( KeSuhC.. n~e oC..C..Ur~ f:,v-

Svbs \.~rv r;~ lvil-o (120 ) w e... o b Y-c;t V"\

k II I-..!. =- I ~ . t . k.l. < 0 ,) I< ti <. 0 . ( n .. 3 )

( 12. I )

Q.23) c.~ "' b~ solve.ct 6.'.J v s.\ "'.5 ( A2.1 1) lo s.v.bsh l v \-e... f'ov- K11 , K.l.. '"Tl, v s

_J_ = I o<. l. ( 1 - 0' +- f?'- ) I - o<. i.. (I - p..'._ '1. )(I - ~'~2.. )

S.o)"~.,,.) fov­.(' re&y ve..V'\ c,,i .e_ S)

w

,_ r I )

cl... (I- f\ ~- > I ( I - (> ~~ )( I - ~ ~ ~ )

(124-)

,..., "-'+ w_ (w.,7.. - W+CJ-)

c.JP ~ + W_ 1.. ( I +- '-'-+ ) G...1-

ov- l [ 1. l.. c CJ+ ) ] '- c..;p +~_ I+ c.J_

Th e. <..e.ro s o.f K .1. q re ~, "~II\ b:l CJo, ~CJ+ VJ_(w e,__+-W+G..J_)

. lJ ..... '2.. " +- (,J a.-.o(

I/ • \\ I - L' f c.J < (A) <CJ W <. CJ < G.J "- J.. wl IOC2,.. \r"l.e2)0-..Y1Ve,.. ov- + 01 ~ - o~

(,.}, ~c..>" ... -:>.re the resonance frequencies. Their exis tence is governed by' whethe r the last inequality of 124 is satisfied. In fact we can plot the loc i . of the resonance frequencies with increasing plasma frequency.

---------r:;::::::------T'T-----------------,--~~i---------------------">C..J

<'----- w 0 , l.oc..vs, \

\

--------------------!C.J~+ ________ _:::,."-------U)--_______ .::::::~~-.:..,~,:...-,.;t_~(,.J ....... , ..,.,,. ~

Note that for CJp<(.,.J+ no resonances occur,for W+ <wp <w_ one resonance occurs,for e-i,.">w_>c..i+2 resonances occur. In the h i gh density lirrit u~ >::> c->_ >>cJ"i" the resonances become ·Jw .... c;.J_

(electron ion hybrid resonance) and (,.Jp/.r:f (isotropic plasma resonance) discussed in §3.12 The physical significance of the reson8nce frequencies,is that waves incident on a plasma at such critical frPquencies will be absorbed. The same analysis can be carried out for other geometries and various direc tions of the static magnetic field with respect to the propaga­tion vector (in particular transverse magnetization) ,but such analyses are usually quite involved and offer no fundamentally new ideas. Besides,longitudinal magnetization is by far the most common in experiments and in the ionsphere. The results for the resonant frequencies of the surface waves hold for any geornetry,and are thus an inhPrent property of the longitudi­nally magnetized plasma.This will be taken into account later,when considering interactions of chc- r ged particles and oscillators -~ith such a plasma.

J. 14 QUASI STATIC METHOD FOR stow WAVES

For waves propagating near the resonance frequency f->--"> oo This is effectively equivalent to assuming that Vpi,,~sL/c.. ~o ~ o.- l"'-a.r c. is infinite compared with the phase velocity. I In fact the electronic thermal veloc ity is a lower bound to the phase velocity) This means that fields are essentially static,and hence B can be ignored.It must be noted that the above approximation holds only if both,the longitudinal and transverse wave numbers are la~9e . This is true for plasmas whose transverse dimensions are small compared with the free space wavelength near the resonance. Quasi static equations for surface waves.

y~ ~ = 0 "3>- .j '-f> S.t. § =: - '!_!.{'

'Y._ . !2 = 0 =9 ~ •( E 0 \(. • Y tf )= 0

As~vVY1·'"'.3 e..x1>(-Y~) ole.peV1deV\c..e, wa... f;vid rhe. e.q,.vC\r .• o .... ~ fo~ ~ lo"'~~Yvolinc..!l~ MG. 3 v-i &·, ze_d p (CA..$VV\.~ ro be..

V'~ <f.._ -t- p:t 'f i.: 0 E~ ~ ~ "f ~ E.. ~ = - V 'fl. T ,) " J -T -I

Ovr.s'ol e, ~e i=:>le:.s\'\l\.£A we. ~ewe.

V7x E 0=-0 \7 . ~ E 0 =0

/ - 0 -

r?l../Jo_ 'l. o t:::- 0 -v'f 0

VT I ·- ~ '( :::::. 0 ) ! - o

(r :i 9)_

i "'s\~e. plei..s""'"

( 129)

I

I I I I

I I I I

Bov"' clo..v-_j Co"' J.i ho r"I~

~~ ( ~~- ~ o ) =0 °'"'c:A V"I •

S v rCo.c..e. wa.ve.s cxv-e.. cAes.c.."·:1 'bed lo_:i J ? x. f,._ .X. ~

y = A o P~ x. E = A i e ,._ E -= - A e ><-e w \, e \re <l-P -~ f> J ~ = ? x "- / ~ o 1 ~ 0 K..L

l. o o -Rx. -i> o · - {?><- E - A 0 r2 - f-'>t. V\ free S\)C\CL Cf = A e. ,- , t::~ == A J ~ <2. J :>L - ,_,e. .

... A0 (3> = -Ap,c.. K.i. ' P..,0 13' ~ -::-. Aj 0' c~Y-0\N\ bou ~o<.C\Yj oono\.1\--ioVts)

H e. V\C. L rs. == - K .l. f }(.. = - k j k I\ r<. .1.. K...L I_,

\(11 k".l.. :=..\ °'~ci ~ >0 . C"-'e. IQ.~ev- i""'fl i e..s I<, , \<: 11 < 0 ) · Th is again yields the resonance freq uency deduced in 3· \3 However,t he quasi- s t at i c method cannot predict the dispers ion r elat i on, because its i nherent a s s umptions do not take into account the manner in wh i c h t h e transverse wave numbers inc r ease near r esonance - an import ant factor determining the fre quency dependence of the propagation constant. Th i s i s evident in the analysis of f i g . 47 Quas i-stat ic method pred i c t s a bac kward wave near resonance ,whilst the correct a na lysis i n J .12 shows that as r~~ a forward wa ve is observed . J.15 ELECTROSTAT I C SURFACE WAVES (?.

7)

As an appli cation of the quas i static me thod,consider e lectros tatic wave s on a magnet i zed olasma(variab le d irect ion of B ). As ment ioned pr ev i ously these wav es are observed neaP. t h e plas ma r es onance frequency r...:>~/.r;: ~

;

x, ~:)

$#.rWff11,Y;#,t;fl111 .,.,_ . '\ e ~~=="',..,,.,,,~-7 ~

4j

Resolv ing the components of ~ along the

[

k .L ::.-. .. " e,...1<.,cos2 e (k ,,-1::.j_)c; ; .... e(.A;>.s.e-

K :::. (K,, - l<.J. ) ~"" e c.os~ K.j_ t..0S1. e + Kii ~i."' >..S-

- - K,,.. !>' "' e- - \<:. x (.,o ~ e

x,y,and z axe s

le:.." s\ ... e 1 k,. case

K..1..

(1 ~\)

4 ~ . 2 K .::: I - & k ::: I - ~ \< = d_Wb Wp

II (;.J.,_ J .L e.Jl.-c...:>b>- ) x c,..i(c,..,"'--f:..Jp1.)

For the e lec tros t at i c field we have ~ = -9~

\l · D=O ¥>~ -(~·S:'f)::o . (\32.) ¢

I5~c~\"'.:> h ""'e. cl.e f:>~v-.de"'c..e.. C"- ~5vl'\.'\e ~ = -'?/)<? (- \<. )l.) :x. ":>0

'f l :: 'f ( -,t ) 0x Vl ( K ~ ';} • ;- k "b ~ ) )<_ < 0 .

Sv\:.!> \.- ~ Yv \:-~ "'5 foV"" 'ft. . '1 \'\.. ( \ °3.2.) w-e.. f;.,__J. ( .(o..r e.... hOl,N.OSG?..V\e.OvS p\~~YIAG..) K o-i.ct • o'f~

x.. )(.. - ~ _i k K - -+ ( k "2. 1< K 1.. v ) lL' .: ch....... (} £)" ;ll.a u ,~ a 'd a -\- ~ .... 1' 1r 1. =- o Mcx\.-d-.\"'.j r<:\.""~ev-.l<<A.\ §. _.f\elds we... obv-C\.;V\.

'f ( ")l) :::: ~ f' ( x, ">L. + j -x.." ~) (1~4)

I I _I_ f ( l. ~ 1.. t~ w.,e.-ve -x," °""' kd K,..'a- /1<.lLx.. C1.v..ct -x. = 11<:.c...x.\ ""Jlx \<.) k:~'a .... ~ K.n ... ) - 1<1 ..... :>l~

Since x is negative ins ide t he plasma ,i t is c le a r t ha t the f i eld : wi l l decay exponent i a lly away fr om t he boundary exe cuting os cill a tions of s pat ial period i:rr/)/.'. th e ~ e os c i llat ions wil l be absent if~ -,oJ k~ =o

0 .,. .p:o~<li) &=-O _,\iii) e-""'~. Equating norma l components of .Q at the bdry.

(1 35)

40

~e pa.rating real and imaginary components cind using k~ /") = cos q,; we... l-iavc...

~~V\,; J K.i..s\"'"'2..e c,.o<i.1 ¢ ~ k.L ku (s.i"''l.¢ +C,o!.:i...~LoS 2S = I k:)( \ s 'cv1 ec.o.s<P - 1 ( 15' )

Consider high frequency w8ves lelectron motion on l y ) 0quaring both sides of (136)and so l ving for <,..>

W = i [wb si~e c.os4' :t J~wP'"-t-c..>p'1..C I+ ~·'"'1..c:P+co.s.l-4;>eo~'l...e-J (\37)

For~ lo"'-~}\rvd\ V\ "- \ wQ.V'!. (&-=~= ry"l...) cv = 2 -tJ~:+c...J ... 1..) =Jc..;p ..... +c,.J'a.... (1~9) ::l....

(o._<;. ·,"' °"'3 · 1'3 ) In general this equation correct solutions may be origina l field equations CRITICAL ANGLE PHENOlVIENA

yields spurious solutions. rhe number of determined by substitution into the similar to J.13

(a) f or B parallel to boundary (t7 = '){) e-<i-td w.,, < 1 surface wav e oropagation is pos~ible only for si.._4>~wb/(..)p c..Jp

(b) for ~ .L ~o o.vi~ '-'b/c;..ip..,., surface wave propaeat ion is possible only for c ll)s ~ ..:::. (;.Jp/w b

1hese critical angle effec t s are studi ed thoroughly in the next section § J.2 EFFECTS OF AN OSCILLATOR MOVING CLOSE TO A PLAS~~

GENEEAL CONSIDERATIONS } I "' ""s Y-e_5Y \rlAIN\(.... \-1,,e,.. osc.AllO\.toV' . \,-,o.S

I I W'IOWle.V\\.-S ~o ) ~"' a.111o( tv-~Ve.111".J w,,'

'X Is wav<LY'lv""'b~ ·,.., ~L 1> ol\v-<-c.N-:o~(= \(~)

~ ~ 62,.M ,._ .

The system can be analyzed as a sum of Four i er components . Define equivalent electric and magnetic moment densities

Pw =(P0!+iiv} e-jYX ~('a)((~ -~) ) W\c..J =(W1°/ 41T\T) ~jvx...S('O) ~(~-d)

\.J~ev-e. 0\ b (~'a) , b 0 [ kd ) o.~ Fo u~·,e...- c...o~; e,;e,_"' \.-.s .

C\o(\c'a } -=- [ ~ L ~ · folJ~ - €;, µ, W/c_ [ 1:: · ~ol 1

b~ ( k 0 ) ~ [ ~ [ ~ · ~o]] 1r + w/[ [ ~ · £,., J 1

} ( 14 0)

A\.so ~ = (-vJ kd, - -X.. 1 >!j~L"), - ol.)) v ::::. J G.J2. .... ".l... ( T k p ~ o."'J M_ Q -=-0 ) ivi ";>€,r,-v - Jc~ "'"" f<> =- ·!.

We can write similar equations for E etc.substituting the correct Fourier coefficients and correct dielectric BOUNDAh '{ CONDI'l' IONS

V'· D ::::.: VI • • 'Y >< ~ = \/ . B. = "' . v .... H = 0 - - -- - - -Tuc.se, ...-e.d \J C..<2.- ro

8 H<..2.) 0

for constants.

';) ( 0 (J) H < . ..1) H 8-> o'b Hc.J?> + H w~ ) - _w\, fa,(Hw + )= 01 . ) \ w'b }A ;i_ (.,.)>-

8< _l i..) } (1+2) () ( E 0 + E I.I) ) - E ( o - (j) r\..2.) SL. 1::::w I b(..J T c ) = E~ ()1 -w'& w2i - 02> }. .) 1J wl> -C..>1

f.ubs \,; \.-.,y.;"'S -f VOW'\ (_1 40 ) &. (14-1) fo.r \--t, <?_. {\ C?Jd~

a 0 + 0 1 G. i. IC, I (C..o+Cl 1 ) :::Ef.4..2... } -Jl , 'X.~ :J (14~)

6 0 + 61 =- ...!::>~ )'\, ( 60 ;-- b,) = j'l z.. bz. ~,

)C..L

Sol "i V\3 yow- 0.. J b

C\' ( kd} =

b, Uc-a)

Foy

\..l.J Ee..>

:.

""'X, I € 2. - 'X-2. &,

~rEL -T ")(.2- ~I

-x, \ )J. l.. - ")(.L. )A I

"').'..I ~~ ~ ?(.,__ )J.1

l °'-o I

bo ,}

-·a~\ (~.,_J.) -~ \(:J '}

"b

E (l.) (..J'b

b<u..o\N\e­

E "X.~ - X'­E 1'~+')(.~

e. j e olkct

c1 +r;) I

l\.) ~ h:i.. Viol S f ov- ·, ~ ~ ol. e.&.1. V w Ow Q. ( SV VVI o~ o S cA I l °" l-o r \,./ o. vc..

Y e.cyu·, v-e...d. ~-r CC""\- ", \I\ \Ji lt~ .

• (°t.) := - j'VPo - dv'c.. (o. ( e..-~ )'...,f ),- cl l + Xt_ -x_.,_ H(...J - e... J ,_ X· +)l 1> .+l\ \TC,. -o.'> L L

t-l} -~ Y.C soc j Ka ';f 1' j ~~ - ) )c., o4_ H = - ! Q p. e e. ''- '<l- ol.. \:: 1-

(..J JJ .2:nyc. -oO ~i.t-'t-2. (_1+~)

E11STENGE OF SURFAGE WAVES Surface wav es wi l l occur if EX~+~&:O i n order that a po l e exist in the integrand of E"' . Also ""t1.• '>l; < o s o ~ hat t~e wa.v ~s are i evanescent from the in~ ~rface .Thus we obtain a dispersion relat_on for s ur f ace waves -x.~' 1e.1:i.. := x,,_2.. Svl.os.\ri \rvh:. fo v "Xi, I x'L

CJ '2.. '2.. ..... "L . ·. (c..J >c:. '1. - ,, "2.. _ k 6

2 ) 1 e 12.. = z,_ 1~ 1 - -Y - ~ ~

.' . (16 1 ~ - I )(-vz.4- K 2-) - W~2.. \~\( IE: \ 1- IJ = 0 (14-7) ~

Co~s;dev ~c... llYo..vt... hvW\bev- v-~ sokJ \.-- ', o~ ol~c....~""°'""' , .., """'<- i"'l-ev-~ce. p\c..t-ie. ~

Fi,3 5'1.

Y-::: I<. .J. C..O s Cf > k °d = K ..L s.'1 i.... 't - ·>rt-a. .... Cf

U s.i ll'I~ ~e_se... v-e..!o..Y.:o"' s (14-7) ~\"""pl i '-i e ~ \--o

( (b 1-1 ) >' i.. ~(. 1 'f = w 2. I ~ \ <=""'-

!:> u b s \r~ 'r v h \t\.J f ov- Y::. (C,.J :!: W 0 )/-ir ~ o_.,.,d sol\J~V\~ for G...J .J we. hC1..vfL

Wo

(....> ::. I \ :+ n.. j '~ I \~ (£\ - I

( 148)

This is nothing more tha.n the l!~~pler formula for surface waves. 'l'he + sign i s a pp l ica.b le in the anoma l us cas e v->c..i,;. which cannot be realized in fre e space , and hencP wi l l be ignored hereafter . Equat ion (1 4B ) can ther. be rewritten as

GJ J IE.t (ltl -1 ) (1 +c;)

Le.r ~ -w0 )/~C<r.>~-== u.,C..w) a~cl

Plo\'b~s ~. (w) & <A-~ (w) we_ c.o-11\

l..l2.. ( w )

) 1€! (;..) (€\ -\

f '"'"' ~e.

= t.-L<- (w)

s.ol...ihoV\.

J.21 ChITICAL ANGLE~ ¥ram the above graph it is evident thatbac~ward wave solutions will always exist ,whils t forward waves e:xist only if '( > 'f.._:"(where 'f...,.· • .,., is deter mined from the point of tangency of u,and u~) If is worth while to investigate the behav iour of this critical point as afunction of plasma parameters.

(A)FREE PLASMA "J. L

From A pp end ix II we have E = I - c..;P /oJ CJ "l..

Since w<GJv for surface wave propagation \El== ~ ...... -\

..u..t wp/c.i = X- .. It\: ::Li_, L<--lr [>Gc>::>lf = b

T°h €.AA (I .cf-~ ) b lZ...(..O ~ e.. S

(I !>O) 1-0.~

b 11-, e ex\ h c..o-. \ po·

1"' \- \ ~ c1' e.v t.v """·,"'ed. b~ .e°\-' v o.. 1-·. l"\5 1-1\e... d ev-·1 v0t.. Fr ve s

o.\- bo~ S\ des . • . .e. - C\ ( i. - .., - 2.. ) I G >!-... ~ v t. J ~ = ::>c. ~ ~ :x.. + 1 I l( ~ .,_ - i.) ( -x.... -l )

~vbs\--·, Y-vh ns foy- b fvol-v\ (_1 So) Ci.1-1d S'i'-""'pl~f~\V\_s 'lL + - 'l. )(.... l. - ~ + 2 == 0 o r ( o.. ";)(.. + - 2..<A ;,t, l. - ')L r l.c.. = ~ )

( 15'1)

This equation '2-an be s olved by using a power series method or by numer ical analysis on the )200 computer . Both methods were tried during this project but are toolengthy to present here. In any case we can use an approximation shown by computer to be accurate to 1%. Since surface waves will be observed for Wo << wP

(close to the plasma resonance ~damping will be excessive .) a must be very small.Then (152)reduces tm

::x..~ - ~ = 0 G"-

Subs \.-i h .1 r-·1 ~s

~~·'"" -= Cos-'

( No\'"e. ~o..."°

'""' Q S"O) we. ~ ' V'o(

[ ~ ( I - c._ ;ih ) J w C...0-.\1\"' ol.- lo.e Y-oo

"'

b = c...-

; oV\.-.evwise ' lOV\

k ~ vp).

- I , .

(B) LON~ TTUDINALL'l MAGNETIZED PLASMA In this case the olasma beco · t · of the problem become~ too d~~~-an~tsoHropie, and a ge~eral treatmen~ the s f icu • owever we can investigate high ~rf~~~dwave~ wh~ch arise under certain limiting conditions

be revised . F~~rc~~~in~~f~e~~yn~;~~ih~ ~~u~~:~yh~~~di~io:g)hav~,;o Th th - ""-J.. <. T E = 0

us e surface wave condition is modified 0

l "X~ K-1. ~ 'X4.: 0 o r X;_1..l 1<- \2.= 'X l..

:X..2,must be determined fro th d · J. · ,~ ( 1 5+)

magnetized plasma from A~piieChisp~rsitohn relati?nfor longitudinallv problem in hand · • anging e notation co suit the

p ~ = kd :2.. + X:. 2.,i.. y l.. = - y l- K 2... =- CJ i. ol -== 1-c. ~ p., -= ..t X-

( .t = Wb/c.-Jp Is ~e.. (ll"'-o.._!>11\e.Y;c ~~e!cl)/oteYs.~Y-"1 ) u •• ._) vO.v-io..ble...)

4-3

App. 1f l... ~ ':>t.

k.:: \-:x_.'l.. I - :i<. K =I-" _, K. .l '= I-{. 'l.::ic....._ ' .,... I+-..(.')(.

Rec°' \Ii\'\~ \(

Kt..= , _

""e.. o\:spttv-s\0V1 Y-Q.J~ho"' be.c..o\Nle..~

c, 4 _ [ -Y ic K.11 _ \) + C..J 1. ( I< 1

+ K.,... k:'t. ~ 1<.J... c... ~ I t::

J... Covis\ol.<t..v- 4- \i """'·,\--;~ CC\.se..s

T L <<\ IC?.. c wb <<c.Jo << c,..;f>) wl.-.i~ v-e_prtie.t.'\rs 1"'1e. low !-'<?...'°' ~i~h ole.V'lstY_j li""'~r . T1'e..V'\ ~If \~e k's be..<.o""'e. \ -.:>c..2- 1_q_ ""'e.. P' °' ~""'c.. 6e.c..o"""es iso h--o p\c cu. t "'- ( A ) .

II l.-==- I J >~ ">>I (cz.( w b = c.....>p ">> c.....>0

) . \"'h~ 5 ~olo(s -fo...- o(,~ I Jl-~

pl<A SW\e:...S ·l"" ~ode.v-CA.h~ .. ~~e.Jd..~ (s.'l:J O . 03 webe,y IM-l-)

~ " = - x. 'l... ' K .J.... =- .:t , ~"'"" -::. - x. )

(l S' ' ) \:, L<A:>""' e. S

+ ...,2..( '- wi.) :i.. p - _>--- y -~-~ p -4 c...

.)(., )_ ( + CJ+ 'l...) -- 0 - ')) -_")C.. 2...- c..4

['\2. - '.:)t.'1. [ · '2.. ...., \.. r - - y ~ .:.W

'r --z;: -+ j-y+ 6 Q'- i.. w+ J - - - "Y -t--

c.~ c..."'"

( ""t.e,\eck·,~ \-evVV\5 of (1 5' +) ""Ow b €....UV""' ~ S

4( c..J?.. l... ) \.. ~ - \<a - JI l.. ==-

0 ~ o( e..v- '/".:>t. l.. )

J ~e... \-e.,v-~ i"'- sqva.ve... b~c..ke..\.-s ~vSV vc. "'·,sh .

15 +

~e.. !.ol v 1-- \ 01'\. ~ ~ .s l.A i JaJol ~ Y + + !::l.._ ':J..-'- = 0 ...J c4

( c..o~s.\de,Vi~ pro(>C\,_y:t\.-\~ wa.ve5. o~!.J i-e . Q_e.(w)-=/:O)

~e...c..,,Y ViO w b..e...GalN'I e....S

w :: I - r-- J;Z

n \ .s I s. ~ e... I 0 w fve.cy-ve.V\ C._'.:1 ~ Y"""VVl0\~) ""'e... ~is"" -!.-e.q,.ve..vi c~ Ccl.se ( w ~ GJp /\,}'2...-) ,· s ro o ol ~ ff ic..v I 1-- t-o o..""-a..1~ :z..e .

Ill .l-:>"> I This ~olol!. -\ov- VE'Ar,j re.V\vOvs pl~.svvic..s I\'\ i"'l-e~.se

k I\ .::.. I - )l... '\.. ,) K. ..l.. ::. I ) l<-t. = I-

')(._

"I) k ::: r ( ~ ~

J...

p~ - [- Y \. c~-)l'\..) "'t-c;..;l. ( \.. ')C.. L )]p"l..+ .. - .\.-~ -<..."- ~~

( l -~\.)(-Y'L-t- ~~(1+~)).>c.

( 160)

Go"' ~\d e..v- (a.) Yt-.e... h ~ 9 ~ f v-eryv e""c.j ~5.e. c..;o ~ Wp/J"'" ,_ ~~I .

P+ - (i..-){\.J( - 'Y?.~ ~:-) p\. + ( 1- x,_ )C-Y"L+ ~~) L ~o

PL. == (\ - 7- l...) (- ")/ ?..+ ~2. ) OY- p 't..= - ')l"L. .+- =:_'l... c...~ ~~

T~e two solut i ons f or ~ are e ncourag i ng for as s hown i n App I V an a n i sotrop ic p lasma a dmi ts a superposition of t wo surfac e wav e mode s f or each mode i n f re e space . Con s i der each wav e separate l y

(' '') (I ~2.)

(i) (c.:_""'l.-yt.-k. 0'1..) = (1-x.'-)(-"Y?..+ ':~) - kd~ (S-\t.4L . WC\ve. e.q_ .)

l - ~ (163)

(ii ) w i II SC\ I· is{~ wQ.ve i{'

<O (t6+ )

(1 6 b )

Hence t he wav e i s a supe r posit i on ( each wi t h a cr i t i cal value of 'f )

of modes of va r ious frequenc i es

Not e tha t ""r >CJ > _ w_ '°- - - :--<2- I I - f1:> 6 c.:r> ~ \

(16{, )

This i s a pec uliar wave wh i ch i s c omplet ely symmet r i cal about the plasma boundar y .It seems to me that its ex i st ence can be caused only by t h e pres ence of instab i lit ies in the boundar y l ayer . Now c onsider t he low f r e quency l i mit -< ~ x ":>'> 1

p.... :X.. '?. ( 'Y 2. - CJ 2. ) p .,_ ~ )<.. .,_ (-v t. + 2 W 't ) y L =. 0 c.. "- c: '-

To o. '.Jc o ol. °' ppv-o .xi~ &\. h O ii\

'2. ~ l. ) . p .=: 0 Ov- \D 2.. =: "'X.. 'L. ( 'V 1- - _ . (., ..,_

Sv'bs \-- ~ \rvh"'~ iV\. (IS+) WCl- f- \ "'cA ~I- loo~ wo..~;. a..- ~ ~~ SG\ VVie

Y -:: (,.} ::--..... (,.) 0 c... =:./ CJ =-

1 _ 0-- i.e. . OYl.e_ 1.-)0..v e.. \ s pv-e...Se.V\\r.

S UJ.\!TIVIARY OF SURFACE WAVE EFFECTS IN I NTENSE MAGNETIC FIELDS (J.J

F . 53 15 . POI NTS OF INTEREST

(a ) zeros and poles of K~may s i gnify a n abs ence of t h e Eyfie l d in the wave , and regions near thes e s i ngular ities may exhibit peculiar charac t er i stics .

(b) waves of fr equencies .c lose t o ~ and J~.,_w_ (t h e free plas ma r es onanc es) wil l be seve r ely da mpe d out . I n s hort t he spectrum of r a d i a t ed sur f ace waves will be comolicat ed.

3 . 22ANGULAR DISTRIBUTI ON OF SUR FA CB WAVE ENERGY" For s imp l i city cons ider only th e unmagnet ized plas ma . The actua l s urf ace wave f ie l ds ca n be evaluated by th e method of residues a s the poles i n (145) & 046) Afte r s i mplify i ng one must convert the vari ab l e of i ntegration fFom ~toX~and subs t i t ute i n ( 147) E ll) "l-1 \ ";I_ - y.

GJ = -po ye,..; k,I [CV.._ I f<.. l l.. 1-/,IK IL. 1)~ "1.. ) 'l. J ---;_ II - y ll II - I j )(

.l.c 1T {k1\- I (ll<.11

\ '--1) c.

X <'..l<f> ~ - "-' ( pal ) _ J [ ~~ll<n\ -Yi(lknl - l)J "\';j \ '-' J I)(...,, I - I LI )( ,, I - \ J y"-

t; l l ) = p 0 Ye..>.,_ I l'-11 \ 2.. t ]- \~ cu~ [ ~ (k1d - v.,_(\K,,\ -\) J4..

:J.vc.,'L J\1<,,( - 1 (I~ i.-t) c -z-

x ~p ~ _ CJ( \K 11\ ~ - ct ) c J IK11 I - I

1. ]~ - i L z'-1-K"l - y"-(\\<:"~- '> 1) \ -.lvx_} " [ \ k. l \ \ - 11, ~

The important thing to note is tha t the H are both zero since t 11eir integrand s contain no surface wave poles . Al l the fiel d s are c oncentrated close to the s u rfacepeffectivelywv,rithin ~ 1 = ~ J xi. - 2 (CL'oovQ. bol"'.'j) & ~ ~ ~ Sc.; J ~:--~ (.bel O\N bcl ¥'"~ ') •

The x,y fi e ld compon"'.!nts c a n b e calculated fr om "\1a xwell,,s. e q u a tions a n d substituted in t h eP o-jnting vectoroA fter l a borious s i !Ilplific:a tion we o b taintthe a ng·1lar power d istribution

dW Poz.CVo4- ~4- (ir ~p [-c ~wo d J~7-'f of'f cl x.. + 3 (1 ) J J1~ .. 1- 1 I\ - p....-.LJr>tf \

V-c. \~11 \-1 <fw I' - r--n W-::>\f \ 5

Compa re t hi s with the ener g y d istribution of an oscillator moviing

Lhrough a d ielectr i c

~ ::::. p () "l. w 4- !IV s Tl $ '1 "'?, "f _J l.j) ) d ~ I ( Lu~dG\V &' L'1~sh'1 f2, (17o) x.. 4- ~c... 3 o / I - ~"' ~Cf I s-

rne two distribut i ons a r e quite a iffrent oThe or d i n a ry d iel e ctric exhibits a strong ener gy p eak a bout r--,,.~<f ~ I , wh ilst in t h e p l as.na this peak is d a mped out by a n e x po.iential exp ( -""/S) where £'~ \f. .... ~f-1\

(is very s inall)0A lso in the p l a s ma backward r a ;.; iation is a lways present,

but in th e d ielectric for war d ~:: o ano b a ck\va rd \f oe.If r ay s a re '.'li ' sing . A ppl i c a tion:= The ma in a pplic a tion of this type o f ana lysis is in ionospheric physics (of e a rth and s t ars) wh ere uist u r Danc e s · s uch a s elec t ric o r mzigne tidst o r ms or fl a r e s c an p r·.i p a6a t e a.lun g ti1 e i onospher i c b o und a ries , a0u c a n ce ei 1 ecti v ~ l y rep l a c e d by os ci l l a tor s.

§3 . J I NTERACT I ON OF AN E~ECTRON BEAM WITH A PLASMA VIA SURFACE WAVES

Vlo ) c c. ~ .._. 'V. '> ""<>"' o e vie..""5 e.,'r- '. c c..-1 ec..Y.-'P"' be.a.""' ( o\e."'s ·,t-_j

l/////l!l/1/l!l!ll!fl//i//!/lllT/JJJ//llJJ//J/ fl//fll/////I/ -· - l' I "'-SYY"lc.. ( tL-\ e..U-v-oVl d ev' ~~ t~ No )

Co vis~der f ie,,ld vo.¥"i~bles D( ~r z~ [ I<(;) d- + kb} - est J 1 Now = o- ( ~.,( ! ~.. H f.. )

re btZ.- el.Ylo

(o- i ~ o( e_ \rer~·,V\12.ct fvt:> VI-\ Bo\ t".z r>-ttA1-t..,'5

-1,1-Wl..L = ~o ( \ - (So J ~ - --- --

Wl.l.(c..i -Kb 110 )

w-.e.. S$ / f> o := ) ( 171 \

- ' l. J - J !l, n,. X- n;, ( ) ( E )L - {?>o H ~ \

.1. t...l - Kiv.. o )

• 'l.

~ :::. d e.., "'0

[ CV (_ I R. 4 ) E 3 ~..L. (lJ - ~ -,}" )2. - l -'° ~ 't-

( M~ \c\ V\5 U .se o.+ 1-h<i- ecy vo .. J ·;o \I'\ of c,Qll\r: vi J \-; V.j ) .

Thcz.. 12..Cyvo.\~~'°Vl~ ·,V\ (111') ea."" \ae.. S.ubs,r-;h.Jt-~J \V'l\ro HCA x weJl's

~uo..\~ ; 0V'tS > o.."'cl SO \0<2..d fo,,,. rv-->o r.:j pQ....S' ~ wa..ve__s. . O) ~1 =0" (n) H1 =-0

To s~Ls,\j tt..e.. plo_S\111\CL bou\l'\do.v.'.:l 001Ad~ V-~OV\.5 o.... ~vpe.x-pos.\ h ' o"" D~ ~e. Se.. V\rt io o( e_ '::. ~ v :s. \.-- be- u ~e.d.

}

( 17,1.. /

")t. "-

H )(., == A ~ ( c ' a. K'd + j c2YI~ X-_)e-x x.

H = A ~ (!' c, a.. :X. _ -c, "}> k:<l )e.. -d

1 -J

(Tv-a.visvevse. C...OW\ p 01r1e.I" \--~ ~ev-~ve..ol a. s. pYQ.v~o1...1s.b) .

-X.-t- :::. J ka 'l...-+- kb.._ - ~~ K 11 ~ ( -X-+- ) "> O ( 173)

A\.- H, ~ p IQ >Ma.. - loea. WI i V'I ~e...rta-c.e.. I= :>(.' H b ) 1-1 ~ a. v e... COV\ I-; h u 0 u> ,J

bvY Hj ~~~CA. d\s.c...oVlhvtvir~ ( w~icJ..- CO...Vl loe.. e.vol..;o..re.d, b:j ;"'~e -

dro..H""9 cv-r~(!:i) o.rovV1c1 !-'he.. bouV'ld.a.:J. Appl_j;VIS )-t-.,e~e.

(.oV\qlr:onS we.. ob\raiVl h-H1 ... 6iSf>CZ...v-~',0"' ye_\o...\-;oY'\ ~V" s=-v"'fa.c..e..

wo..ve'::7. Fo.....- \ow d.<2.V"S; 'r_j e...le..c.J-.....--oV\ foLD.W'\S fti.i ~ S.i\IV\rli fies. f-o

X, + + Ji(.!. X _ "::: J:: _ VJ;., [ "1;; ( -X + + X _ ) - ( ~ i- + \< ..1.. 'X _ ) J (I 7 + J (n-b:z..-1)( X+ + X._ )( w - tc

6 "Vo) 4

(NoY<Z- ~C\.y- fov- ho e..le...c.JrroV\. b~C\.VVl , (1 7f-) ""e...olvc..e.S 'ho

X: + k x =- 0 + J. - - ~S IV\ Se...c.ki OV\ S · 2. 0>'1 1'1..-,e. C>Sc.-( lfa..Y-or,)

4- if e. 2 1"1 0 .J W'I h = h1o ( I - f-' o l.. )-~

s;l.. 1...-=- 4-11 N 0 e 4

p ""'

( .Lso \-v-op\c.. pltA.S.vY\"'-) > w~\

Io w be.ct YA d <2.n s i lr_j

X+- + K ..1.. x _ := 0 =;> ( I -'- ( S2.. L) C.V Z..

) - \<.. - I- - -c,,.,; 'a- c. :z...

e./--::: c...>l..(k) = [K2 c1.. + ~;] ± J ~e~ -r k:+c..+ ( l. 'L "-)

K = K(S + K~

Fo~ St.P >">Kc. ( H;gh de..V>s..·, ~~ .p\~S.V11\o... or- low wo..ve..\e...M~~) SvY\C\.(.<2- wQ.ve... ~v-<2-~ve,Vlue..~ 6~<..o"""e..

:t..

w (k.) = 1- k. c.. ( \ - -:::: ~1... ) --~ l<c. st.. .....

A 're;v-- vio.. l-\ve..l.:l ~ov- <;l..,o r'"r ~o..ve.,S (117)

c.v (. \'-. ) '::: + SL p ( I - Sl.; ) -=;:- S.'"1.. VJ""7.... Vi.. g \.('2.c, 'L

The.... pv-e.S,12...\/\C.cz.- of O.V\ o...pfre..~I (fi.b\e.. be.o...VVL C..Q.vt be. tVt'rvoolvce-d.

\ V\ 'ro ~<2- Q bo v...e... C2._cy\A.O- ~~ O ~s. b...:J WO._j of pe..r l'vr- 60... h'o ~ . ( 174)

wi ( l V'IOW _j ·, a.Jd CA) ~ k~ v o ( l -t- ~ ) tr> l.-i e_y-e_.

~ = + l x_ u,.,~ [ "'h"L ( -x..+- + ~-) - ( x.,.. -r \<..L ?c._ )

c...> ~ ( IA~ l.._ -I ) ( -X. ;- -r X. _ ) ( ?l +- + l<.i_ 'X _ )

0

(179)

(17q)

:. $ 2<'0 """'S ~<l... w<Av..'- VJ.1\( be.. o... sp°'-LGl.1~ ~v-ow~ V\.5 OV) e_

.J

w'•lrh o.. S v-ew~ ~"e... tf::: I""" ( w) = I<-~"'" o I""" ( £ ) -Sv'.osv<\-..;\ .... ~""-j fov- ~e. v~v-\o..b\es l"'- (17&) ~e... ~~""°"

'6 cit... w I ~ ) v'2-\ /'10

lh e., ~ y-0 w ~ Ya_ \-.c., b e. (.,o W"l €. _s ?°'-V" \,-; CV l ~ V' I~ t C\. v-5 .e..

l.J ~ k~ \) c ~ c..; (. K )

(181)

Tl,eV\ we ho. v~

£ = -\ -t- ~ ~ 2

'"Th~ VVIO...XiW\v~ ~v-01.-> ~ vcd-e. , ~ pv-0pov-ho""-a.( \--o '?:.~ .

A similar analysis may be carried out for a beam with velocity spread but it is quite complicated,and its only new result is that all quantities must be reolaced bv their statistical averages(by integratioc J.31 AMPLIFICATION AND GENERATION OF SUBMILLIMETRE WAVES BY A

RELATIVISTIC ELECTKON BEAM The above principles can be used to 1esign new and superiour amplifiers (the conventional resonat ors and slow wave guides §2.09 are cumbersome and difficult to design,because their dimensions must be~ ;w < I0-

3"'-)Instead,we can pass a strongly relativistic electron beam near a plasma,so that asurface wave is excited W-:::. kc.. Ji - (~c:&.)~ ~ Kc..( l - -k (W\~c:"")~) ~ k-Vc (I !3)

The amplitude of the excited wave differs appreciably from zero over a distance A_~~- from the boundary.We can control~- at will e.g.For 10 MeV electons A._=+..,,...., ,whereas ""- =- ?\/2.1t =- o . 2. .,,..,...., . The growth rate at the surface wav~ frequency can be obtained by substitution k.c./SL. = ~<> c. ~/ E

p

){wio.x = ~ [ ~~ 2. W (~-Ji;- ] y 5 (1 8+) This is very favourable,but unfortunately requires a rather large beam current.Experiments carried out on beam-plasma system?(42) show a maximum gain of 2dB/cm. near the resonance frequency . This is rather disappointing but perhaps could be improved upon by modifying the dimensions of the apparatus.In fact there is a limit to the growth of waves in a plasma,caused by self-interaction

(collisional damping)and inhomogeneity(present in most practical systems). These are described be low.

§ J .4 INTERAC1IION OF PLASM.A WAVES (d)consider a sinusoidal unmodulated wave in a homogeneous isotropic

plasma with low collision frequency (_say c..v = 10 3 "))) .

Le..~ E = £ .:> eo s ( c.J t - ~ · ~ ) =: E0 ~ lf .

n ~ £. \ ""'~ \\ + \ e.d d .:1 n °'" ""'·, c ~ u °" 'r-i o"' be. (,()V\I'\ e .s c( '\T., - VI ;::::; • C..O

~ O..t = e.. !::o t-0":> 1 - ';> "\Tl = .e... ~ s.1i.- f l'\I\ Go..:>

( e le..ch,·oVI "e..I oc.< 11-.j ') . -d .... -.~r v~lo~\r~ .

~

We.,, how.e... -v "2. -iJ" ~ 1.1 ""2. +- 2-v, V-0 ~fr :: ., '

1- v a..._ .rz.. << v, "' 2 '"IT, \)" 0 ~ l.f ( S,; VK.e,.. 0

y o< """° ( ~; VY\ p \ 12.. \ : Vl ~ ..- c...o H i s; ~ o "" \-1\ e.o v-~ )

.6.: = .6 v _ A \J-o = [ e ~o ~ V\ 'f J 2..

er \J"" o Ji: V\l"I v-o w

1iic:. c.1'CA~5<2- '"' y CA111 Q.ff~c..X- """"~ valve.s 0 .q ~~ d ,' eJ e .. c .... h·-·tC... rQ..VI S-0 \'"' CO\l'\;\pO \I'\ e....11\ ~s . (IC II & k. ...l. e.l-c.. ) .

'L "\ll )

( 185)

I

I I I I I I I I

p ...-o-c.J-ic...cx.I .e.,)( ~ h'\? \e... Co vis \ ~ e. \""" w "" ~ \-ts I c. V"l 2.. • -n... e."' E :-~ ~ -::. 102 w y

~e. wcx.ve .

K ;::: ,, . I-

olO..Mp\V':, w\ \\

(b) Wcwe...s 1 V\

If w ~ 'o ( i_ ~

This

0 c...c,..;>V .

o-.. wo.ve::

;(..W~IO+ €0

C~VY'~\ V'l_5 ~ Y'--' V"' '-- "

s vb s \-i vu r; ':.5 _., w.e. ·P "'°' )) ~ l 000 -V . c:<...S c.. r-eS. u IV- of

\S

€ = k ,, -= \-\'\'lc,J ( w - d y )

Ne...~\e.c.N ·,n5 1o\I'\ &Mor'i o"' , f:::: e.. AN .

.·. 6 E. = - 4ne.-i.. AN e... f: \'v\ (,..) ( c..J - J y )

H 2N'lc.e.... J

(c) Wcxv<Ll.

~ O V\ 0 VI i \o v ~; \.-~ ""'a.~ c\l!>o ct. ss:,s lr o{ °'-VIA P' "'-.5

\\I\ ~OV\l\05<2Me.ou"E> .> Clll\\so~rop.1c.. p\e\.S>V\.'\.~ · .

"""e.. e..,\l i ? r~ ~I\~ po lt\.v-\ z e.d ex. rv-aOV'd t V) O-.rJ1 C,o "" s ·, J. ex

E == ~ 0

s-; Y\ <f + E c.os 'f - (E. -'°!:> )

y,1\-,e_V'Q.. &~) e\:) C..V'(!.. C\.~.,9\e,S be .. h.ve_.e,,\1\ ( ~~()..) ~0~ C...l;\d. k

G<:>vv,pO""e,0Y5 o~ ~~C2.Acls o.\o"'!:) ~ Cl.Ye..

~ <.f E ~ & 0-:r:> <f E o~ Gr.> cr"' ~"' ; '°1.. b

New) ~e, ~W\'('oh'2-ll\\.- o~ ~ . \::: CL l o "'~ '< / ~ -k. ( E 0 c.. Gr> & ~ s\h_ ~ + E 0 '!> c....r.. e b V:r-.. tf )

.' · ~ e :::: (_d. iv ~ ) /4 ir - - ~ ( Eo c,.:r-::. & o... <;;.i... ~ + ~ 0 b. c.>-::. &._ ~ ~) 4-ll 0... ';>

~:- = -+c;e: N ( Eo<A-~ e 0... ~"' 'f T £0 ... G>-:>eb ~ ce ) .

n e.- ~ 0 ""5 Q, I V\ ~ e,., p-€.N'llvl ·, H- ~" i \r_J t-'0tr. s 0 r , · c;. pv--o po r i--: 0 V\~ (

ro ~e._ eyva.1,\ ~ ~ ~j 0.., b 0 V/2.... .

For B strong ( cos e.._ ,,G.::>seb)~I ,and the disturbance will be more pronounced.Effect (a ) is quadratic w'-"' (,..), ± l.c-.> ,vvhilst tb) & (c ) are linear wl -"'> c.>' ±w . In a real, experimental plasma all these thr ee must be considered,thus rendering a proper analysis difficult. However,it can be noted that in principle they are analogous te scattering and interference of w~ves with phase shift and amplitude damping.

§J.5 HARMONIC ANALYSIS Small pulse propagation through plasmas can be treated by using modif i ed transformations of the type described in §2.12. ( Qualitative results are presented in (44)) . In practical terms, i=!ignals with appreciable Fourier components of the o~~er of Np will be severely distorted by the highly dispersive plasma (with the possibility of backward waves ) .This could be a serious problem in the design of a succesful surface wave aerial to overcome the spacecraft reentry blackout. (The rocket's surface and reentry plasma sheet are analogous to the system studied on p.36 ) .

Recently (3months ago) a new method was developed to solve for surface wave transien+,~ in a plasma (4 5) .Rather than using transforma­tions,the authors carried out an elegant,but rather lengthy

( therefore omitted)Fourier analysis of the Boltzmann equation. Unfortunately due to lack of time the full value of this work cannot be estimated.

y9NCLUSION

From this paper,and literature cited~it is clear that surface

waves are a common occurrence in nature.Their nractical applications

are widespread and vgried,and warrant an investigation in their

own right.There is great scope for research especially in

connection with plasmas for use in such divarse fields as

controlled thermonuclear fusion,and space research by radio­

astronomy.

Al

APPENDIX_L_ F,~ ld Ov-1-ho90~0 \; ~~ Pv-ogexhes Orl-ho90V)o.li~ of .surfQCQ. 'W 0." ct. W"l0 de. s.. Cons; d«.r a. c..~ lil'lch-ical

x.. I + - ~ "..-- - -, E == 01"1 .§ ..... -;- t: I \ I l"I - l't. a>O I I 1-l } I ""' L. ci" ~ .,-+- H+

' , ~ - +

) " - It.

I (A 101)

E L_ b"' ~ ..... - ~; } j = '"+-'l-· I

' c:.Y1.INt>t~AT 00 "\i " 2> <. 0 / _ l}l .. - - H :. L_ b.._ H"' -t- ~ R. "'

W "'e\'"e. ~ R & \j R. o.v-e. olve. \-o ~e v- o. ot ·, °' h' ovi f; e.lo4 > a v.d §""" & I:!., a.v-e_ ol ve. 1-o

H-.e. S.vrf'o.c:e. wova.. ~ode. o\" ov--dev- V'I. Tv-all'5vev-Se. svv-fqCe. wave fie,/o\s cu-c. '.­-+-

~"' t = ~VI e.)(.p [- ........ <. 1 E.,.-..._ = ~"' t..)(p [. .. r.., -i l H + = 4,. ex.-.. r-ri <l H---' = -~ ... exp(:-f'.,.J > 0 , - ,---..;: 0 , - "",-f: - '""' r to'\ O , _ \").~ 0

Tue 1-oro..1 ~•e.\cl \S o. Sol-.1HcV1 o~ 'Y_x ~ =-jw~ > ~')<ti= ~WE:.§' 4-- ;! ~ w\,;tls.1- rh~ SvrfQ(.e,_ wo.ve. YVIOd<2. is o. solv\-;o"' of' \-(,,~ ~cv.,..C.e..-fr-e"l e..~ vo. h'ov.s,.

B.':I 1-1--e l-ov-~"'\':z. Rf!..C.:1p..-oci Y:::J (..ohd;\-;oVI ( A I02)

g:p$ ( § )(ti I - ~I X lj) '~ ol Q : s ~) ;[ . §I d v ~ I\ §.) t! de.,,ol-e.. 1-'he VI""" svrfa.c.e ... wo.ve. .,,..oole, Qvid. § ,) !:L 1-\-,e_""" we.. hCIVC..

2' • ( §' i< I:', - §°1 )(.\::!)·~do.= ?'t:(§: )(.tl,-~1 ><-tj) + 9 1 ~b ·( § x~ 1 - .§',~ ·ti)= 0 ( AI03) _ ·· • y ' ( ~ '1 )( \j V>'I - f n.. )( !:I YI ) = 'Y. t · ( §.., X \:I"" - §°M >< !:\.., ) - ( r" + \-:. ) 9 "} ' ( ~"' ><. t!..., - f "'°' >< 1-J .,)

[ ovie. W"IC\.j how reploc.e. ~ ... , §. ..... , ~.., 1 lj..._ lo~ rl-.c.. 1-.-o.v-sve.v-sc... <...cvnpovH~. lf'll-·-s o""l~ ~; \l'lc:.e... ~e o..x ·, cd f i eJd w i II ~ i e..l d "'o w"""' po"' e"'r o.loV'\ ~ 0 1s- w!..- c...., c.vos.s£ol w'i µ,, c.i.. :1 vC..ctOY-J . He.V>c.<..

sr.s Yt : ( ~h x ti- - S:-- )( 1:1 ... ) al.a ""(r., + r .... ) I f.s. g_ 1. [ § .,,t x !:li.-.,-t - ;""',t"' B ... ,t] o<c.. (A l04)

bu\- J~a. C!) · [ ~..,,t x ~-.t. - ~ ..... -t>< ti ... ,t ]o(o. = ~ ~. ( ~ ... x\j.., - ~-" \j._) cH. ( A 105)

How e.vev- ~ e. <..o"' l-o v v- ·,.., ~ ~ ~ ~·o.I Vo.\/\; s he.S o"' \-1-- ~ c..o.,, ol vci-;"'j gvi de. """°" 11.s ( ~ x. §" = O \ Sul:.s\-:ru\-i"'-,5 yov- ~ .... t:, !:\ ... ,t e...lrc.. we. oblo.i"'

(r.., + r""' ) 5J5

9 ~ · ( g.., x ~"" - ~- x {,.._ ') olc. ::: O ( ~i...,ce. e.x\Jo"'e\-11-\o.I s c..c:.."'ce.I) (A 106)

C-0Vls.1dev- ~~· , ~ .... ' wl-.eve. ~e.~e okpe"'~ o"' e..xp t_r .... 1 l ( D.1 .... c..c.l.-iovi of w.o.5""e -h c.. ~\ e.lc:A ·, s v-<L vev-s~cl ) . ; · (. . ~ ""'• t = - ~ .... e.x ~ [i'. .... 2'] . ~; s ~i e.lc:A.s (A, ~i -w. ·, le.- eq..vo.1-; o.,

(f,.,- r ..,,) Hs <a)·c-~ .... :x~ ... - ~-x~ ... >°'C\ = o CA<o7)

Ad.div'.3 °'"'d. .>vbYv~c...1-~"".:J 12.cy. (A IOb) °'"'°' (Ato7 ) we ob.\--C'\'•"'

s l ~"' x ~ ..., · ~ 1 ~a.. = j [ e,,..,. ~ ~"' · ~ ~ olo.. "'" 0 (A I oS) s J ~ This is ~e... ov-~090V10..\i 1-..':l 1--~ lo..l-io"' . Re.,,\ohoVI Be.rwe€-\ll $y.-'°'aLe.. Wove... Modes ~v.d f1.-e.. Sv?ple.w-e.V\h:~r..::l F ielc4.

Lei- ~ 1 \:I be. K--e \-o\-o..1 f\e..ld & ~., \j, b(!.. s.0.-fc.c.e.. wa.vt.. f\elds:. (of o....-olev- VI ~s bc.fo~ ~e. Lo.-e."'r' i"'l-'1..~v-o..\ vo.,.,ishe.s ove..1r \-1,..,e. c..2ili"'d...-:.c.o../ £.v.--fc;.ce... ( b~'-4v.S.e.. of '"""e.

v-o..cl;a.ho"' fie..lol. 'oe.iv._s \oov,,.,o\e.d. cd- M, 0.1'\cA l~<.. .S.vrfc.c.c.. W<AvE... loc.\"'~ e.vc....,e.S.Ll2V\~)

Jf(~~ ... § .. ++ ~R· )x~._+ - ~ .;-( ~"'--~~ +~~)}~'*~-JJ[(lb..,~;+§~),.~(-~;,.(~b ... H:,.~;JJ~a~ }"l. b>- .,

= J)J .§:·I cJ.v. Afh~v- svhsl-:h•hv.:, fov- ~<!. 4'.iuo\ .s a"'ol -isi"'.5 Ol'"~ o-30"'0..li~ r~I ">'. )} ~= x ~"' ~f\ - ~"' x 8; e-r ... 3Ji.]. ~~a. - Sf[~; x-; ... ~r~1~ I=:\~><~., ~r ... 1r, J . ~ d Cl. == - ))) ~-- - - --., JJ ::::-.., _., ..:~ '

Si...,c.e.. J>, ~ h1. o..-e. av-6-.r v-ctv,J , 11..e. ~!..,.. h.vo \-ev-""s of ~'- \~\- siole. ""'vs'- be. V<>VIS"°'\"'1-.

Howeve.v- ~\v.C<.. ""e. Y"O.d ; 0,,i.-; 0 "' ~;.e,lol Vi~.s. a. C..O\l\Yi"'ve"s e\5e..,vo.\ve !'=.pe.ch·vY\.'\,, ivs:

c\e.~cl~c.e. o.._ ls CA">\.\o\-- Q.. V>"'v\ K,.,~ e-ri., 1,,,1 e.-r'.,1,._ Y-~f>e.c..\ive.IJ.Ht.\.1C.e. lt-.e.. C..011\SVa"'l-5

""'vs.\.-vo.""i~\.\. i.t. rr_:t6 e. ~~.-<:\ o1.~=srrE~ . Jo1v (A110) J J ., - - - ' - 1> l ) - .. - F. cl - we.. ob '-~;"'

f"ev~ov-""'i""5 "-'t... ~o.\Me 1--~pe.. q °''"'a..( ::3~i s ~., .. a. wo..ve. w..o d£.. ~ ~"' t!"' -:t~ ... J5 ~,,,~~ .. ·9~ c::Ao. = )~\ 'S:·J cJ.v. (Alli)

11-.e_..sc. rwo °'2.Cj\JO..Yi0 .... $ will c(e\'el"tM\>'I~ ~<Z,.. Surfc.c_e..,. \,..IQV(.. 0,\.'\<.pl ihJo4eS o. .., C\"'~ \o>\.

~ · a· c.. ~'-'ov--Y. wv"'q_"'"' e...le..wie"'"' <.!>(le"'~""" d.{ ;.s e~vivo..\e..y,\.- \--o o. dipo\e. f , wl...evL

ctc..JP: 'Jclt. Fov- a. s"""~n (.A)vye..11\'\- loo f> ]"~ ('}' ;.s v,,;i, .. rc...ll\5e."'\r) W.t..~C'\v'C!..

~ c.. I ~". '1 cA l ::: r SS ~ )( §:". ~ d ~ = - ~·w )A 0 T 5) ~ .... ~ cl.o. = - j LV rot'"'- ti '• t-.\'\.- J.e CZ. • veal b rucJ.- G S~s,h_.,._ (rj:: IH~,.ol.o.) is ~;~a\L l'\'\O\'VIC2.'-"I" . Tu~ Svv-fc.c..e wo.ve... ~""'-r' v S )H .. • ~ -'.,.

,lQ rr e. )(.~"' . ~'L c:AQ = - ;w E"'-. p + ~w }lo ti:. ~ l. b., ( ( ~ .. )C ~ ..... 0isdc...=- ~(A~:. E ... J.W}"o tk ~ "'J)-"' - o 0

- -1

),) (A\l'l.)

A2

APPEl\JDIX _II

Uy"\de.r >--he v-e.sr...-",c.hovis src~red Oh p.34-, 1-1--e ol_::JnO.VY\.1C eq,va.hon v-e~uc.e..s. re wiolv _ ( ~ - e. §o + ~x ~o ) - \'\-1~ Y ( §o , ~o-~f>pli<d tie\Js ~ ~-c..o\\is.10\.'\frec.i, .)

A -:.>v ..,,. i VI .j e...,cp (jcvt) cl e.pe"" d. '2. V\Ct:..

(Yc..+jw +~bx ) ~ = e~/W\ ( 0Jb =-: ~o) le..r £°E..L b12.. 11--e ve-lo~r.:i ..L re H--e.. cJ.;v-e..c.,hoVI o{ E

V._ ::. l y' ~ + ~· W -t- W ':> >< ) ~..L °\,)'" :::

- E...L [ . M - E,1 Ve.. + J ( w - wb) 1 [_ v ... -t- ,i Cw~ wb)]

IV\ ~e.\rl.sov- fo...-""'

[~TV j ( .t -v) OJ . e. E v = J ( l +v) .t. + v o -=-

0 .2W1

0 ::tp

e.. §11

V Si Y'l:J ~e,_ V V\0

1 h~v~ r...-o..nS~Y-li"'O...\- ; OVI V ~

"'l =

Now T "= Ne.~ =

k -== r~~ ~~ ~ J b 0 kp

(J. E D~f Ivie. ~ = I+ (j1

w&) 0

= [Ki - \<,,

~J liv. ori3i"'a.I <..-oov-di"'o.~e.s. ) - Kx KJ.. 0 0

KJ.. =-(k~ + k,... )/J... } K 11 = kp

Frowi ~"-e. cAe-f;.._;1--:0"' ~\:.eve. we..... oh.ro.iY'I L L

k11

= I - ±-. k = 1- _o<....:__ _ _ I - J y c.. ) (.. I + r-: - A y c:..- J

o{ "L

I<,... = I - . 1- r-~ - ~ 'fc..

h C\V~

(A202)

).

(A 2.0~)

(A ::2.04)

{fi205)

(A 206)

(A :t 07)

( c;1...-::. !::::!.f._ n.. 1

= l.J!> '(/ = ~ w :::::.J"'c.~ = e.\e.c..,\-.-o"' pla.s.m~ .c ..... ~oveV\C.1..4) c..• 1 ~ - (....) c;... c.-J p ""' } -, ..J

M~xwe:,ll'.s e.4v<Xl-~0111S c.c."' v.ow be. v-t..dvcll..d ro ~· ~ : 0 1 Y:' · _f? = 0 , 2 x §" = -~ 0 ~ J \l x \:l = w _e ( Q = E 0 ~ i'" ~ G.J ::

Vsil'l5 ~e pv-opa-5""-ho"' ved-ov- It. ( ~i.e.lo\s ~ exp(-jB'.!:J)

(A :z.og)

t 0 ~ · § )

!S·D = !:·~ =O > ~x£ =C..lB J l<'>C\.j :-w~ (A 2.09)

The. wav~ ~cyvo.hoV\ c.c;.V\ ~vs be.. v-e.duced \-o

~ ~ (~ x § ) - fo 1.. ~ • ~ = O (A :i.1 o ) (ef'e.c..hvcz..\~ 5 e..q,VC'hOnS C....C""-.\:;•Vled ·,"' \-ev..sc-r 1(>.-""'). Fov--"' solvho"' \-o 0xiS.,_

~e.. ~e..'revVV' 1Y1a."'-Y V'\l\\JSY vo..~··s"'-. Th i > ~ie..lols o.. olis..pe-rs'10"' v-e...lo.\.-;o"' ( 4-"" or-cAe.v-iv- "-) \.../hie.A.- ha~ be.e.."' ~ov-ov_s~S ·1n.ve.~h.so..re.d b~ Ash-o""' \..J~o ~olv-<..ol ; \ - -\:cw "Q.rlovs °'"'Sl<.s be.\-\,Jee..it\ ~ ~"'cl t§0 . Sv~ ole.J-o.\ \ed. C\."'c...1'.::ls.is i~ 111~1-...-ec:v"·,v-c.ol ~~~­Noh:. :- 1.f iOV\5 Qve. \·o. l::e.V\ iV1\-o c-.c.c..ouV\r

l< t <= I - °": al+..... - - _,.,o<.._'-_ __ _ l+r-~ 1-(\.' ' \ (H·{)'_)(I-(>~",

Wh e. r(.. o( l-: N <. .._ ( ~ ± -t- w._ )

E.o W .._ ( ""+ \1\1_)

Reso"'~"'ces. ~v..d C..v~offs. .

> K = 1- o1. ' f A 2.1 1)

> II ~

l<e.soV\~VIC.t.S c~c.. u....- ~ov- K...- J Ke~ 00 , Cvro~s. f<:>.- k ... , kt..---===> 0 A'r r~So"" o.""~ ~.e.. e..""eyS~ is C\.k.>ov be.d b.j \"\-\e.. p\c;;..sw. C\, C\V\ol b.Q...low Cv \-off \-\.-.e.. wa.vc. c..e.c..ses. 'ro ~ pv-opo..~CA\-·, ~j . Fv-o ..... ( P\.;l.\I)

K.i = \- ol~(1-(\;(?') I< -= ~oli.( (1>~ -(\') (A~I~) (1-~~ ... )U-('>;") J " (l-{';>~i.)(1 -r<'·)

( K,.. I~ Wa..vL). Fv-oM (A .;tll )a"lcl (A.:l..l:t.) wrz..

W~~~ i \. s~t.,d,~"ide.o\ \\l\YO v-e..5\o"'s. C.Ovr~S.pOV\d~"'.S \ro V Q. ...-\C\JS Y~\)C.<; o...f c..vro~"' C\."'d pv-c~o..5c..\--;V\5 wQ.vH. (Su~ o. pie\- ~or ~e.. ~h-c<.ovd;V\o.v~ WC\ve..

\ S S~OW\I\ OV\ V\fll'l<.Y rCL~e..)

T \.ie. 'Z..Xl-v-o-o re(·, no..~ elli\)~;c.""I~ t-olov'• z ed

C\c..v-o~~ g~ .

.,....esioV\~ ~ -<Q o.llow i::-ro~~~c..r.""'~

./ .__ __ 7""__;:= ::..._ ____ _._ ______ ~ d, .L.

Pl.,ljs.1ca.I si3"'if 1c.c::u."ce.. a.\ ~~ p"-v-""'',\-h vi~~ l-e.."'sol"'

A3

glj det"\\.;,v;o"' 7 we.. ~(\v(.. c.. ~ 2... k (' h I . ~=\/\,.. = ,... ~\-c. W~e..-e. u.,...,u.e.,<A..l" ~v-e. c..o....,y?o11~\'ll-5

O~ p a.S~ vQ... oc.... y::l ( rdo.hJ,.. ro ol.; r~ c..l--i o"' ~ \")Owe.v- -\low - see Appe.\l\d ix ii'f ) l"'hv.s l<r, \(~, K11 ~1--c. o.v~ VIOi"\,,,""~ ~ove ~l\.11\ sqvcn-e. ..-ooh. of \'°1\e..

&\"'i SO\'..-o"lc. ,.e_,v-o..ck\"e 'i"'de.;x.

Wo..ve. i'o IC\v0

1 "2.o..ho"'

w~e."' ~(?_ d.isj>ev.s·10~ rebhoVI is sa.\- i s;'-i<?ol ( de..\-' ( ~1< l$ -\<0 '- ~ ) = o ) we.. ~; ... ~ k--C\.Y E~: E~: E 1. Cl.-L <Z..q,vc...I ro ~(.. ro.1-ics. of ~c.c..\-o.r s.

A~ rev- Sv b.s\.-a....,.r~ c:d \l"e.Avrc.."\j <2.M.f..\'I~,

)( "'t. si"' 6 c..o s. ~-+ I<~ 1.-) +le:',<._, k~&

(A..<13)

E"'-~ ("''--K~'j )(\l\'2..i.1"'1-S-- k~}.) T- k1~.._ > E~ ;0("''\.-1<.J'a

E ~ ~ k~ x. ( "',_ s ii.. "I.. &- - K ];~) + K. ~ ~ l "' ,_ c.o '> e- s i"' & + kx 'ls ) Ori"' pcn·hcu\<=1."', wl..e ..... k~l.= l<i-zL=O Ex...-;: C."-..t-V1"t-)( 1<.,,-"'\..~i..,_l.&)) Eo~ K.x (.._l....!.\"'l.&-~n), t:'b~~'-KJ..)"'L.s·, ... ec.os.&.

APPENDIX I[ EY'leY~j & Pow~v ~o...hOV'l~v- ~e..s. \..-. ~lo..S.IN>a.~.

le.r ..!_'10 =No~ (No ;s v""pe\rl-vv-loeci e.le..c.1-\ro"' olC2."'.s\\.-.'.j) .

Co""se.v-vaY~OV\ e.qua..v;OV\S (fvo"" G.o\~'2..IMC\."'"" ~-) ye.c("c~ ro V'·I1 + ~N = 0 -W\ 'dy +C2..E +cz...'\rx\3

0=0

<>-i: ) Clt - - -( N 1 § 1 I' > \j ex ve.. p<!.v-h.Jv \, eol q..vo...,, r; he. S) . Dor wiv l \- ;pl~\ "',5

- r-i r . <>'.l!' -to- <2... r . E ..,... e.. ( y x B o ) . " =- a - ()-t - - - -

Do,._ MVlhpl~·1n9 M~xwe..ll's eqvQ.\rio"'s w i }'\.. tj J c. ..._J - § v-e...s pe.cl-'1 vca..\""

,,j 'i) ~

\:± · ( ~ x € J ~ - _ro & (H~) .1 - ~ • ( '!_ )(. ~ ) = - £0 E C>t (A ?>03)

SuW'IW'l0•v._s WQ... o\,h~ ·, V"\

V · {. § X; J ::=. - & ( 1: }Ao H '- + -l ~o EL. i- -l Now- "\TL) (A3o'+)

A ssv lf\o\.•V\_5 eACp CJ· wt ) d. epe"'d e,.\f\C.....<2-

'?_ · .£' + J' GJ N =- 0 J - J. w \'YI "J[ + e. § + «a. y >< ~ 0 = 0 .1

(A 305) SJ><~ :::. - ) W ~0 !_-I J Y'~~ = jur.,§. 't C2. _!"

( fcd\r t"evVI-< ·,5 d.1.112.. \'"o a. d(po\e. 'i'°'revc.c.\-;o~ of ~!Z.. ~c..-3~ Th~ \-il"'l.e.. ava.v-C\._yz.d Q.."'e.v-5~ d..e\o'\s\\.-~ Co."" b12- eNcdvC\.h.ct o...<;

W = S l ~ }'\.:>flj\ 1--t- i E:o I~\ '- + i N~\1!11..} o( V (A 307)

Co.,s:1dcz.v \'lie.. voluw.e.. \\l\Y€ ~vc..I o-t 1-1,,,e_ v-'15"-"~·,d~ o{ (A306). I\- i s\-1-. Q..

<...o~t>'ex po\oj(l,V- -\low.> v.>l-ios.ll.... V'4!,""\ pa. .... \- i> \'\..c. Po~"'' ;"'.5 ve.c..1--o ...... W~e."' ~e.se.. rwo C\v-e ~~v a.l ~(... \fC\"VV\12.. \<;. sc.·1d I 0 bQ.. l\I\ re.!.~V'I0..1-\C.e.

E:quivo.l<Z.\l'\rl~) W'l~k.i~ u.Se. of e.~ = Q:" · E C\,\o'\d C.OW\rC\ .... ~VI~ wil'1.-. (_A. 506) I G c_.1'1i \<'. J_ 1.. ~ 0

- . = . § == 4- t 0 Is \ - ~ N""' I ~ I i.. + ±. Ne... ~ ~ ~I §. 0 (A. 3 OS )

~u..s wQ.. CC..VI ~;V>J \"1<Z... hvvoCI.. Qvevo-.5e..ol. ~VI~~ cAeV\~iY.J *°J ()3Cft

W = 5~c}.v J w: t,Mol':i\L+~~ol~ \\_:l N\M\y\"'-= 4rol'=l'-+i~o~ ~~~ .£ De..C'w1Q... w :: r l !Ao \H\)_o\V w ,.,5.l C:o\t\"\...otv WK=5 J..,...;W\ \v\ .... dv

I ""' )" 4 r · - ) e ,, 4- - ' ., 4 -- _\ I W_. ~ -l. N(.. l.£~X -VJ: f" ol.v

<'\ v GJ • :?o .

(A 3oct ) ~ W = vJ...., ~ We + w ,<

At

. · We= \.-le. - W"- + w°' ) We)'.= we.+ w1<:. ~ v-<so"'"'"'c.e. ~ w - w =: 0 "" " .

(A310) Appl~ ~e VO.\r\Q.\'iovic..I pY 01V)c."ple_ \r.o l A 306)

~ ~· (b~ x !:!:¥- + E'"'" .}lj) =: -~w( w .&~ - (wK-wd.) A~ I V ><- v- <' '\ + - t.J e. - R - .l. ' 6 ~o ) ~ c.J

1-\cwi""S clevQ...loped. \~e. v-e.<\tv'iv-e.cl ~C>'"IN\o-lisW'I-> we. C..G\"- ol,,hA\i-- \)owe.v ~low ~cyVI'> .

)°oY- fve.e. °'"'c( 3v\d.ed wo..vC2S. eo ... .sid~v \'\,,(Z_ hH·e,v- <:cJ£ OV\\_J .> bec.o....v~iz.. 'ir w\11 be v.s;:.e."'vl i V\ lc..~t.v C\.\'\c;.1'1~\s . Assvw..e VV'lifcv\IV\.tY..'.:l o-(' 5v\o(e. i"' ~e.. ~ O.i n: .. ckio"' (sc, ~)

et~J. S~~Q.v-etlr"t:.. \"\..\~ c;{.e,v'1ve..J. ecy v ~\ io"'..S \.,yo \-..,.~YISV€N-Se. C\'"'-ol. lo"'ji \--ud.i no..l l'...OVl-lf'<:>"le.111\rs.

-Jl-..lW'IYT +e.._§T + e.. (yxi:So)T =0 - j·WM'V°2 + e..E-i_ + e.(-v><Bo) · i... =0 .J ~ " - - -} o §r . . H " - . . rt - -

O); = Jc.J~),><f"o -T +- ~T t1s ~w r:> H~ + !:_~· y Tx. ~I -0

'dHT:: -~Wi.;, >< E-o§'T + ?r H~ + Le.fx~~ ' a~

~e.sc:.. Y-e.duce \-o

O~T ::. d(.,J ~1 x. }Ao ti T + '!T \;~ ~} · S?'r><~r

OJ> )

d!-:h = -~w !:c) x ~T + S?T \jb- ~ .\l ')(~T o~

-} - T -

i' Jc....> /"o HJ;:. 0

- JCJ .D'J> = 0

T~~ ..- oJ.vce.. P+jQ, =:jA ± ~ ><~~ · ~,,_ J..o.. == ~w-plex power flow

(A ""' c....- oss s12.dr~ OV\ c-f ~ v \ d ~) We.. cbi-u·,V\ ( .\n~W\ ) ~ ( f>+ A&)= - 2jc..J cw.,.. - w€') 02'

} (A 3 \2)

(A 3.1'3.)

For CA. pro f>Cl~O\. y-; o \I\ _:..o"' SI-~ I- 'll : ol. + A (\

'){ ( p + ~ CQ..) = :2.. d. w ( W ""'T - VJ e. "b ~ W \( ~ ) - 2... WV c (A 3 \4)

I~ d... = 0 we... hcw.c.. 0.. pv-OpQ.50\.h"'<::> wo..ve.. -(_ ~e.. Ovtl:i 0"1e. of \l'\Yev-e.S.~)

. . w""'-we.. + w"'-wd -=-O ) w"""-wt.=-o ( A 315)

(A 311)

Lt. = ~ = p /( w.,.,,, - w""l + We..r - We.}J + w te l> - w Kl -t- w~~) (.A 316)

Pt I so 1

u s I V\ ~ (. A 43 \ O ) - yt., ~ v o.. v- °' a. \r .\ o V\ eq u o. \- ; E> "' we.. c lo \ra. \"'

GlW P p (j\ Sl 7 ) C)0- = W W""'+We-t-\,Jic-

1°1-\\J~ we.. ~c.vQ... c.. v-12...\a.ho"' b~rw.e...e....V\ \"'-.Q.. gY-ovp °'-"'cl.. pl--ic...>e ve..lc~Vie.s cic..>_'\I.._ = ~ (w .... ,-w""'

1+ W€. T - We:i.) (A312)

2l r- - pi.-... ~~ r-- u

p~ C, ~-E:. ve..\o~\""~ CA"' "'o.ve._ c~ q>os.'1 re~·,:i ~ ro - W - Wc.1s < o . W c; < 0 °'-~ c( E-r ""

1 c..o"'d. ; 1-',o"'s. Q.YL k111owV\ C\.~ \:,c ... c..kwC\....d. lNa.ve~.

APPENDlX TV

(N~oY a."'°' K.1. _\7T· E_T + ~T · ~....- K_i_ 4-'2T K' ·L- xE - t Li_· \7x.E_- 2SK 11 E \. =0 , x -~ -r ><-e> _,. _, o

IV' a.\\ CA.S.e.S c..c VIS\de-v-e.d \ri e.,v-~ Y' i:::.,. ::. ':l. k.i. = S'.7, K" =. 0 (_ "'o ~ \-n.Je. fc r ~\"' f \ ) Y\1 .S.) the. .e.q.,ua.\-\o"s \'\..e.\I\ s;,""'\')l ~f~ ..-o '\12. E +~5 =bH o.."'°' q'-H;! +cH~ =olE"L ~ ~ 2' 1' ,. c ,i!S 0

1 2. i... K · ~ Kx " 2 k:: i. I<..- K~ W "°'e.v~ a. c:. (IS' + I( 0 K .1. ) -

11 J b = J c..J ~ o o ~ / c.. = o ~ o K .1

~ .1. i

( ~ 4-05)

I • -vXx\<n O'I = -\wE " -0 <> 1'..1...

[} :rYT]e,[;la~]~~[~!I:T]ob~.Md ~ '""~c;;: ~::~3) -U -Q T P L xv H l~X Hr

- } - r J - "' - 'I.. ( i.. .i. )

W'-'e,vc.. p = -'6( ~,_ + l:::,;z. l'-.i.. ) K.,,, ~'-J ,Mo 1'c:o 'I.. K.)( J Q :::. ~ \(~ \( .>< , S =. i:,,_lJ }Ao : + k., K'.1.

C> J:> '4. L ,_ 2-. :z._

I= '1S~'w6-oK.ic I)::: -lc;J~o(ll't.\<..l... + 'Ko.,__l<"K.t.) D= ( '){ -t-l<o K..l.) .,..(ko K_,.) t> .I l) )

For ~e. loV\(J·, \--vd. \V'\C\. ll ~ V\.o\CA..5"' e....H z~ p\ct~""~ (A 4- 0 4- ) C-a\"\ b~ v"'c.ovp lc<.c::l

[vT+ -r(o.+c)V'~ +(uc..- bd)jE1s =O}(A+07 ) [vT4 + <~+c..) V'~ +(a<- - bd) 1 H 1 = O

c l _e_ f'_ (_ · p v- \ we_ ""-vSY ~ewe.. roy- a So vVi OV\ ~ \'\.\e.. )VY\/\-'\ €.h.p - J_ ' -'I )

p+-(o..+c.)p'-T (o..c.- bd)=O

Choose t P t. } s:.t. (V~ - l'.1."l.) E~.i. = 0 l~ =I, 2.)

(A+og)

Tl.-~.s 0

1\""elie.s. E 1 = E1.1-r E1"1.. (A 4- 09)

Fv-oW\ (A405) we. ob\.-0.·1\I\ H~ = .h., E 1>1 + ..f.. 4 E'1.~ w~ev-<1- ~l. = o.... ~p{ ( A+to)

T\.-e £.')(p\ic\\-· fav-\N\ of ( A "\-0~) .> w\.-~~ ~S. ~e. o(.isp~~\Cn y-e.l.,..l· iOV\ i~

p 4 - L~z...( Kn -r\} ;- koz..( l<'.u + Kr-~t. )] p'2.. -r ~1 (l(2..+Jc.0t..l<r)(~2.+K01.K-t)(A 4 11) k.l. ~J. f--_ -

No 're.. l'\,..c>.Y ~ r Y'-1 t.. VV\ WI.a..~~ e-\r; ~ ~ p le,. s ll'Vt~ b = c;\ = O i"' ( A +o 5)

C\v.d K.11::. \(..1.. ) Kr :::- 1:::.(. = .ll.. 1(1\ . ne dis p€N-S\e"' v-elo..v;o"' \:>e.c,o"""e> (A 4-I ~ p ~ = ( '/( 1.. + ~01... Kil ) - id t. V\1-; C..0...\ ro 0.,\.'\ i ~Q\'"VO plC. o( ~~le_(... h .. ; C.. 1-.J

0

1 ~ E. : k. I\ .('frV(. O- ·Vt..)

A PP~Nb\X y_ Vs\"":> (A 4-06) foy- a. C.._':1\~1AcL,.'1c.c..I s,~f>Ye.VV\ we. h~vcz... ( 's~o..-\~ "de.p~nole,.,az.)

E ~ = \ -X AI 1 (rb) - W,M_,,W\ IST (rb)] ; + \ _'!Si. 3.,.,,AI,....lrb)+ ~BT 1(rih\li: - i- L r.... "" r 'L -........ - Lbr.... r ~ -I\\ '.:f1'.

~: = [-_f.._C.K~(-Tb) _ t..J;~wi t> k~(T~:>) l ~ -r[~:~iCK\'k( Tb) -t b~op1<~(Tb)J~ E H \. 0 . <l ..: 0

ere, q,-v~\-~"'j 't' CJ( )iT Q.V"\d ET & E.T WIL ob\.-~\\I\ + J2.4vo-'\--i0Yo1$

fo.,,- A , S. 1 c, D.

-X I' ra... ~ i-I "'f'1 L 0

_ GJ Eo K11 Vl'l I fl '-

iw E-o \:.:11 J' 1:.r'"

fov- .So Iv \.-i ovi S

( _,

K T .L~ I\ f' r~

- (,..) fao""' I"''\....

J_GJ )'lo b ri ....

L

T•

y \<..I w~-:>""' k A T~ ,. "1.-

- ~ ){i-. \<. a'<--J-10 K' B = 0 b,. 'l.. L..1 '\....

~~Q!l2~_l:f _ 1. LOEB,J. "Sur Les Ondes de Surface en Regime Transitoire"

in French E.M.W.T~ Po485

2. VOLMER,F.,J. "Elastic Surface Waves" po1634 (1967) ,see also

p.833,2081 - Proceedings of IEEE, 0969) J. NORTON,KoA. "The Physical Reality of Space and Surface Waves

in the Radiation Field of Radio Antennas" Proceedings IRE_£2_, 9 (1937) P o1192 and 1203

4o WA IT~ J. ''Electromagnetic Surface Waves - Advances in Radio Research (.1964) p.p. 157 - 217

5 o COLLIN, Ro E. "Field Theory of Guided Waves" ----- -McGraw Hill ,(1960)

6. IVIARCUWITZ,N. "On Field Representation in Terms of Leaky Modes or Eigen - modes" IRE 'rrans .AP- 4, po 192-4 July (1956)

7. PALOCZ, "Cerenkov Radiation and Leaky Waves" & OLINER,AoA. Proceedings IEEE l1963) po623

KARBOWIAK,A.Eo "The Elliptic Surface Wave" Brit oJ .of AppoPhys. 5 328 - 35 September (1954)

IRIKOV,

100 BARLOW,Ho Mo & BhOWN ,J.

11. HUh D,R .. A.

12~ GOUBAU,G.

1 3 HANS EN , R .. C o

4o KING,D .. D.

" Single Wire Lines for Surface WavetJ Whose Cross Section has a Complex Shape" Radio Engineering, Electronic Physics (1967) P o424

"Radio Surface Waves" (1962)

"The Propagation of an Electromagnetic Wave Along an Infinite Corrugated Surface" Can.J.of Physics ~ p.727 - J4 December(1954)

"Single-Conductor Surface Wave 111ransmission Lines" Proc.IH~-2..2_ p.619-24 June (1951) also 11 0n the .~xcitation of Surface Wavesn 40 poBb5-8 July (1952)

" "Single-:::>lab Arbitrary-Polarization Surfa.ce-Wave Structure" IrtE Trans. M'l'T-'J p. 11 5- 20 A pri 1 (1 S ')7)

"Properties of Dielectric I mage Lines" IRE Trans. M'l''r - 3 l195?) l'o35-39 and 75-81 Al eo KING 9 DoD.& SCHL~:::iINGER S.P. "Losses in Die le ct ric Image Lines" IRE Trans. MTT- 3 po 31-J? (19 ~'.P)

15. KAO,K.,C. 11 uie l ectric Fibre :::iurface Waveguide for Optical & HOCKAM,G.A. Frequencies" EoM.,WaT. p.441

16. HEMlVIENDINGEh 9 D."A Forgotten Theorem and its Application to & ZUUKER,F.Jo 0urface-Wave Excitation" Trans.IEEE l1970) po132

rtlCHT' .t:!.R,S.L. "Perturbation Analysis of Axially Nonuniform & DIAMAN'l',r, Structures Using Non Linear Phase Progression'' & 0CHL.c0ING.t,rt,S • .P. 'l'r8ns.IEEE ~1967) Po422. See following paper

on p. 431

18. BABLOvv, Ho IVl. "Millimetre Waves and Opt i cal Waves" ~11/lW'l' . p. 389

19. KA.tt.bUnIAK,AcE. "Lasers and Optical Communications Sy stems" E.M. 'lv .To p.419

2 0 o AL L.r..N , J • L •

21 • AlVllTAJ. 'N.

":::iurface Wave coupling Between Ele!Tlents of Large Arrays" pob38 I~~~ A.~. (1965) "~urface Wave ttesonance Effects" Trans. I EEE (19b9) pQ722- 729

,.. E.M.W.T. refers to '"I'l:fE SYTJIPOSIUM ON ELFC11ROMAGNETIC

WAVE THEORY" Edited by JeBROWN

2£::o CHU,& KuUYOUiVl'f'.i.1-lN,hoG. "Diffraction of Surface Waves"Trans.IEEE

240

KOOY,C.

Ant & Prop (1 962) Po 679 also p. 159 ( 196?)

"Surface Wave Antennas-flush mounted an Aircraft, beam tilt angle compensation'' Trans .. I~~~ (1968) Po 1J5-1j6

"Guided Complex Waves on .Slow Wave ..Periodec Structures 11 E~M~W.'l.·. po4b7

'~ Note on the Propagation of a Pulse by Surface Wav e Modes" E.IV!. \L'L p .. 49'(

260 KAnr,0.N.&Kt1..rdu,,F.CoJr. 11Phenomenological 'J.'heory of Multirnode ;::;urface Wave wtructures" E.M.W. '1'. P o also ~ymposium on Quasi Optics

27. VAN de HUL~T

28.

AHORA,.K.K.

''Light Scat tering by Small .rarticles"

,J .wiley (1957) '''l'heorie der Bev9v\"I~ an der Kugel unter

II ) Berucksichtigllng der Kriechwelle" (I.,.G-e,..-..,,..~"' Ann.der Physil~ .. (1954) p .,253 (see also (19?~))

"Unsymetrica.l Radial 0urface Waves" 'l'rans.IEh~ A.P. (19t6) p .. 797

- _;O. BhAG1'J.i. 9 rlI.F.& L;uLL.t<;N,A~1.L. 11Surface Wave Research at Sheffield" ~ GiLL~~P IE ,~oF.F. Trans.IEEE AoP. 0958) & S'l'lH~J...l<'v._~u,FoA. Special Supplement p.219

31.. SNYDER,A.W.& KYHL,R.L. "Surface Mode Propagation A long an Array

SNYDER ,A. W o

JJ.

BROWN ,Jo

J . CULLEN ,A .L.

4. RICH 9 G.J.,

5o RICE, SoO.

6. WISE, W.H .

of Dielectric Rods with all Elements Excited Identically'' TransoIEEE AoP.(1966) p.510

11Surfac e :Mode Propagating A long a Tapered Dielectric Rod" Trans. IEEE (1965) p.,821

"On the Theory of Shielded Surface Waves" Trans. IEEE M.ToT. 15 (1967) p.410

ADDITIONAL REFEBENCES.

"The Type of Wave which may exist near

If

a Guiding Surface" Proo.IEEE 100 pt. iii 36 3-5 (1953)

ELECTROMAGNETIC WAVES IN ST·RAT IFIED MED IA" pere;ammon 1962 "'Ehe Excitation of Plane Surface Waves" Proo. IEEE-1.Q.1, IV p.225-34 (1954) also 104 , part C p .. 472-4

"The Launching of a Plane Surface Wave" Proc. IEE (1955) Po237

"Series for the Wave Function of a Radiating Dipole at the Earth's Surface" Bell Systems Tech.Journal p.101

'~he Physical Reality of Zenneck's Surface Wave" Bell Systems TechoJournal

SECTION III ------------34. CHAKHABhOTI, B. "Mutual Helationship between longitudival,

surface,and Transverse Wave s at the Boundary of a .t'lasma" Vestnik Moskovsko go Universit eta,Fizika no.':;I pp.30-J8(196q)