Plane Wave Propagation and Reflection

8/14/2019 Plane Wave Propagation and Reflection

1/12

3Plane W a v e Pro pa g a tio n

and Ref lect ionR. Jackson

of Electr ical andCom puter Engineering,Universi ty o f Houston,Houston, T exas, USA

3 .1 I n t r o d u c t i o n . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . 51 33 . 2 B a s ic P r o p e r t i e s o f a P l a n e W a v e . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . 51 33.2.1 Definition of a P lane Wave3 .3 P r o p a g a t i o n o f a H o m o g e n e o u s P l a n e W a v e . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. 5 1 63.3.1 Wavelength,Phase Velocity, and Grou p Velocity 3.3.2 D epth o f Penetration(Skin Depth) 3.3.3 Sum mary of Hom ogeneo us Plane Wave Properties 3.3.4 Polarization3 . 4 P l a n e W a v e R e f l e c t i o n a n d T r a n s m i s s i o n . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . 51 9 3.4.1 Transverse Electric and Transverse Magn etic Decom position 3.4.2 Transverse EquivalentNetwork 3.4.3 Special Case: Two-RegionProblem 3.4.40rthogonality3 . 5 E x a m p l e : R e f l e c t io n o f a n R H C P W a v e . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . 52 2

3.5.1 Solut ionR e f e r e n c e s . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . 5 2 4

. 1 I n t r o d u c t i o nl a n e w a v e s a re t h e s i m p l e s t s o l u t i o n o f M a x w e l l' s e q u a t i o n s i n h o m o g e n e o u s r e g i o n o f s p a ce , s u c h a s f r ee s p ac e ( v a c u u m ) .n s p i t e o f t h e ir s i m p l ic i ty , p l a n e w a v e s h a v e p l a y e d a n i m p o r -a n t r o l e th r o u g h o u t t h e d e v e l o p m e n t o f e le c tr o m a g n et i cs ,t a r t in g f r o m t h e t i m e o f t h e e ar li e st r a d i o t r a n s m i s s i o n sh r o u g h t h e d e v e l o p m e n t o f m o d e r n c o m m u n i c a t i o n s s y s te m s .l a n e w a v e s a r e i m p o r t a n t f o r s e v e r a l r e a s o n s . F i r s t , t h e f a r -ie ld r a d i a t io n f r o m a n y tr a n s m i t t i n g a n t e n n a h a s t h e c h a r a c -e r is t ic s o f a p l a n e w a v e s u f f i c ie n t l y f a r f r o m t h e a n t e n n a . T h en c o m i n g w a v e f i e l d i m p i n g i n g o n a r e c e i v i n g a n t e n n a c a nh e r e f o r e u s u a l ly b e a p p r o x i m a t e d a s a p l a n e w a v e . S e c o n d ,h e e x a c t f i el d r a d i a t e d b y a n y s o u r c e i n a r e g i o n o f s p ac e c a ne c o n s t r u c t e d i n t e r m s o f a c o n t i n u o u s s p e c t r u m o f p la n ea v e s v i a t h e F o u r i e r t r a n s f o r m . U n d e r s t a n d i n g t h e n a t u r e o f

p l a n e w a v e s i s t h u s i m p o r t a n t f o r u n d e r s t a n d i n g b o t h t h e f a r -f ie ld a n d t h e e x a c t r a d i a t i o n f r o m s o u r ce s .

T h e t h e o r y o f p la n e w a v e r e f l e c ti o n f r o m l a y e re d m e d i a i sa l s o a w e l l - d e v e l o p e d a r e a , a n d r e l a t i v e l y s i m p l e e x p r e s s i o n ss u ff i ce f o r u n d e r s t a n d i n g r e f le c t i o n a n d t r a n s m i s s i o n e f fe c ts

h e n l a ye r s a r e p r es e n t . P r o b l e m s i n v o l v i n g re f l e ct i o n s f r o mCopyright 2005 by Academic Press.

ll rights of reproduction in any form reserved.

t h e e a r t h o r s e a , f o r e x a m p l e , a r e e as i l y t r e a t e d u s i n g p l a n ew a v e t h e o r y . E v e n w h e n t h e i n c i d e n t w a v e f r o n t i s a c t u a l l ys p h e r ic a l i n s h ap e , a s f r o m a t r a n s m i t t i n g a n t e n n a , p l a n e w a v et h e o r y m a y o f t e n b e a p p r o x i m a t e l y u s e d w i t h a c c u r a t e r e su l ts .

T h i s c h a p t e r ' s d i s c u s s i o n a s s u m e s a r t i c l e t h a t t h e r e g i o n s o fi n t er e s t a re h o m o g e n e o u s ( t h e m a t e r ia l p r o p e r t i e s a r e c o n s t a n t )a n d i s o t r o p i c , w h i c h c o v e r s m o s t c a s es o f p r a c t i c a l i n t e r es t .

3 .2 B a s i c P r o p e r t i e s o f a P l a n e W a v e3.2 .1 De f ini t ion o f a Plane WaveT h e m o s t g e n e r a l d e f i n i t io n o f a p l a n e w a v e i s a n e l e c t r o m a g -n e t i c f ie ld h a v i n g t h e f o r m :

E = E0q~(x, y , z) (3 .1)a n d

H = H0q~(x, y , z) . (3 .2)I n e q u a t i o n s 3 . 1 a n d 3 . 2 , E 0 a n d H 0 a r e c o n s t a n t v e c t o r s , a n dt h e w a v e f u n c t i o n ~ i s d e f i n e d a s :

51 3


2/12

14O (x, y , z ) = e - j(kxx+kry+kzz)

e - J k ' r

k = &kx + ~,ky + "2kz = [3 - j ar = ~ c x + ~ y + i:z .

e 't isa nd supp r e s se d .) The v e c to r E0 de f ine s t he po la r i z a-

f t he p l a ne w a ve . The r e a l a nd im a g ina r y pa r t s o f t henum be r ve c to r k de f ine t he pha se ve c to r f l a n d a n

e c to r a . The pha se ve c to r ha s un i t s o f r a d i anse t e r a nd g ives t he d i r e c t ion o f m os t r a p id pha se c ha nge ,

he a t t e n ua t ion v e c to r ha s un i t s o f ne pe r s pe r m e te rg ives t he d i r e c t ion o f m o s t r a p id a t t e nua t ion . The m a gn i -o f t he pha se ve c to r g ives t he p ha se c ha nge pe r un i t l e ng th

the pha se ve c to r, a nd the m a gn i tud e o fe c to r.

W a v e N u m b e r V e ct orn a homogeneous loss less space , a p lane wave must sa t i s fy

' s e qua t ions , w h ic h in t he t im e - ha r m on ic f o r m a r e t he1961):

V x H = j m s E .V x E = - j o a I x H .

V . E = 0 .V . H = 0 .

D a v i d R . J a c k s o nx7 x (x7 x E ) - k2 E = 0, (3.11)(3.3)

wh ere k is the (poss ib ly com plex) w a v e n u m b e r de f ine d by :(3.4) k = k ' - j k " = t o x / ~ , ( 3 .1 2 )( 3 .5 ) w i th t he squa r e r oo t c hose n so tha t k l ie s i n the f ou r th qua d -

r a n t o n the c o m ple x p l a ne. The de f in i t i on o f t he ve c to rLaplac ian i s:

V 2 E ~ V ( V " E ) - - V X ( V X E ) . (3.13)Th e fac t tha t th e d ivergence of the e lec t r ic fie ld i s ze ro for at i m e - h a r m o n i c f i e l d i n a h o m o g e n e o u s r e g i o n ( H a r r i n g t o n ,1961 ) r e su l ts i n t he ve c to r H e lm ho l t z e qua t ion :

x 7 2 E + k 2 E = O. (3.14 )In rec tangula r coordina tes , the vec tor Laplac ian i s expressedas:

V 2 E = & V 2 E x q - ~/V2Ey -}- ~V2Ez. (3.15)H e nc e , a l l th r e e r e c t a ngu la r c o m p one n t s o f t he e l ec t ri c fi e ldsa ti s fy t he s c a la r H e lm ho l t z e qua t ion in a hom oge ne ou sregion:

V2%I ~- k2xI = 0. (3.16)(3.6)(3 .7) Subst i tu t ing equa t ion 3 .3 in to equa t io n 3 .16 g ives the resul t:(3.8)(3.9 ) k2 + k2 + k2 = k2

I n the se e qua t ions , 8c a nd i a r e t he pe r m i t t i v i t y a nd pe r m e a b i l -i ty of the space. Fo r free space, 8 = 80 an d Ix = IX0,w he re Ix0 isdefined as 4q-r x 1 0 - 7 H /m , a nd ~ 0 i s de t e r m ine d f r om thede f ine d ve loc i ty o f l igh t ( o r a ny p l a ne w a ve ) i n a va c uu m ( H a y t ,1989): c = 2.99792 458 x 108 m /s ec beca use c = 1/ex/-gT~.This g ives the app rox ima te v a lue 80 = 8 .85418781762039X 1 0 - 1 2 F / m . A l o s s y m e d i u m c a n b e m o d e l e d u s in g a c o m p l e xe f fe c t ive pe r m i t t i v i t y , a c c oun t ing f o r c o ndu c t ion los s a nd /o rpo la r i z a t ion lo s s ( H a r r ing to n , 1961 ). T he c om p le x e f fe c tiveperm it t iv i t y is expressed as:

8 = 8 - y . ( 3 .10 )

o r

I n t h i s e qu a t ion , 8 i s t he c om ple x pe r m i t t i v i t y o f t he m a te r i a l,a c c oun t ing f o r po l a r i z a t ion lo s s ( i f a ny ) , a nd g i s t he c on duc -t i v i t y o f t he m e d ium . H e nc e f o r th , 8e = 8 ' - j s " w i l l be de -no te d a s 8 f o r s im p l i c i t y . The r e f o r e , e qua t ions ( 3 . 6 ) - ( 3 . 9 )wi l l r emain va l id in th e genera l lossy case.

Ta k ing the c u r l o f e qua t ion 3 .7 a nd the n subs t i t u t ing inequa t ion 3 .6 g ives the vec tor wave equa t ion:

k . k = k 2 . (3.17)Th i s is t he s e pa r a t io n e q ua t io n tha t r e la t es t he c om pon e n t s o fthe w a ve num be r ve c to r k de f ine d in e qua t ion 3 . 4 . N o te t ha tthe t e r m on the l e f t s i de o f e qua t ion 3 .17 i s no t i n ge ne r a l equa lto [kl 2 , because k ma y be comp lex .O r t h o g o n a l i t y o f Ve c to r s a n d I m p e d a n c e R e l a t i o n sO the r f unda m e n ta l r e l a t ions f o r a p l a ne w a ve m a y be f ou nd b ysubst i tu t ing equa t ions 3 .1 and 3 .2 in to Maxwel l ' s equ a t ions 3 .6thro ug h 3 .9 . Not ing tha t ~7 --+ - j k for a p lan e wave , Maxwel l ' se qua t ions r e du c e to :

k x H = - r o s E . ( 3 .18 )k x E = t o b~ H . ( 3 .1 9 )

k. E = 0 . (3 .20)k . H = O . ( 3 . 2 1 )


3/12

Plane W ave Propagation an d Reflection 5 1 5a t i o n s 3 . 18 a n d 3 . 19 e a c h i m p l y t h a t E . H = 0 , a n d t o -

, t h e y a l s o i m p l y t h a t k - E = 0 a n d k . H = 0 ; t h a t i s , a l lk, E, a n d H a re m u t u a l l y o r t h o g o n a l . A n o t h e r

e s t in g p r o p e r t y t h a t i s t r u e f o r a n y p l a n e w a v e, w h i c h m a yd i r e c t l y f r o m e q u a t i o n s 3 . 18 , o r 3 . 1 9, i s t h a t :

E . E = q 2 H . H , ( 3. 22 )i s t h e i n t r i n s i c i m p e d a n c e o f t h e s p a ce ( p o s s ib l y

q = ~ / ~ . ( 3 . 23 )

h e p r i n c i p l e b r a n c h o f t h e s q u a r e r o o t i s c h o s e n s o t h a t th ee a l p a r t o f q i s n o n - n e g a t i v e . F o r a v a c u u m , t h e i n t r i n s i cm p e d a n c e i s o f t e n d e n o t e d a s 1"10 a n d h a s a v a l u e o f a p p r o x i -

a t e l y 3 7 6 . 7 3 03 1 ~.Power FlowT h e c o m p l e x P o y n t i n g v e c t o r f o r a p l a n e w a v e , g i v i n g t h ec o m p l e x p o w e r f l ow , is ( a s s u m i n g p e a k n o t a t i o n f o r p h a s o r s ) :

1S = - E x H * . ( 3 . 24 )2U s i n g e q u a t i o n 3 . 1 9 , t h e P o y n t i n g v e c t o r f o r a p l a n e w a v e c a nb e w r i t t e n a s :

1S = ]* 1 2 E 0 x (k * x E 0 ) . (3 . 2 5 )2cotx*U s i n g t h e t r i p l e p r o d u c t r u l e A x ( B x C ) = ( A . C )B - ( A . B ) C , t h i s c a n b e r e w r i t t e n a s :

1 1,121E012k , 1S = 2 o ~ p ~ - 2 c op t~ l* 1 2 (E 0 k*)E o. ( 3 . 2 6 )T h e s e c o n d t e r m i n e q u a t i o n 3 .2 6 is n o t a l w a y s z e r o f o r a n a r b i -t r a r y p l a n e w a v e , e v e n t h o u g h E 0 k = 0 , s i n c e k m a y b e c o m -p l e x i n t h e m o s t g e n e r a l c a s e. I f e i t h e r k o r E 0 i s p r o p o r t i o n a l t oa re a l v e c t o r ( i .e ., a ll o f t h e c o m p o n e n t s o f t h e v e c t o r h a v e t h es a m e p h a s e a n g le ) , t h e n i t is ea s il y d e m o n s t r a t e d t h a t t h e s e c o n dt e r m v a n i sh e s . I n t h i s c a s e, t h e P o y n t i n g v e c t o r b e c o m e s :

1S = l~121E012k * . (3 .2 7)2mix*I f t h e m e d i u m i s a l s o l os s l es s ( I x i s r e a l ), t h e t i m e - a v e r a g ep o w e r f l o w ( c o m i n g f r o m t h e r e a l p a r t o f th e c o m p l e x P o y n t -i n g v e c t o r ) i s i n t h e d i r e c t i o n o f t h e p h a s e v e c t o r / 3 . A s i m i l a rd e r i v a t io n , c a s ti n g t h e P o y n t i n g v e c t o r i n t e r m s o f t h e H 0

e c t o r , y i e l d s :1

S = 1 ~ [2 1 n 0 J 2 k , (3 . 2 8 )2cog

p r o v i d e d e i t h e r k o r H 0 i s p r o p o r t i o n a l t o a r ea l v e c t o r . I f t h er e g i o n i s l o s s le s s ( 8 is re a l ) , t h e t i m e - a v e r a g e p o w e r f l o w i s t h e ni n t h e d i r e c t i o n o f t h e p h a s e v e c t o r . H e n c e , f o r a l o s sl e s s r e g i o n ,t h e t i m e - a v e r a g e p o w e r f l o w i s i n t h e d i r e c t io n o f th e p h a s ev e c t o r / 3 i f o n e o f t h e t h r e e v e c t o r s k , E 0 , o r H 0 i s p r o p o r t i o n a lt o a r e a l v ec t o r . I n m a n y p r a c t i c a l c a s e s o f i n t e re s t , o n e o f th et h r e e v e c t o r s w i l l b e p r o p o r t i o n a l t o a r ea l v e c t o r , a n d h e n c e , t h ec o n c l u s i o n w i l l b e v a l i d . H o w e v e r , i t i s a l w a y s p o s s i b l e t o f i n de x c e p t i o n s , e v e n f o r f r e e s p a c e . O n e s u c h e x a m p l e i s t h e p l a n ew a v e d e f i n e d b y t h e v e c t o r s k = ( 1 , j , 1 ) , E 0 = ( 2 , 1 - 2 - j ) ,a n d H 0 = ( 1 / ( c o ~ ) ) ( - 2 j , 4 + j , 1 - 2 j ) a t a f r e q u e n c y co = c( k0 = 1 ) . T h i s p l a n e w a v e s a t is f i e s M a x w e l l ' s e q u a t i o n s 3 . 1 8t h r o u g h 3 . 2 1 . F o r t h i s p l a n e w a v e , h o w e v e r , t h e v e c t o r s / 3 , a ,a n d t h e p o w e r - f l o w v e c t o r p = R e ( S ) a r e a ll i n d i f f e re n t d i r e c -t i o n s b e c a u s e t h e p o w e r f l o w is i n t h e d i r e c t i o n o f t h e v e c t o r( 5 ,0 ,4 ) . O n e m u s t t h e n b e c a r e f u l t o d e f i n e w h a t i s m e a n t b y t h e" d i r e c t io n o f p r o p a g a t i o n " f o r s u c h a p la n e w a v e .

Direction AnglesO n e w a y t o c h a r a c t e r iz e a p l a n e w a v e is t h r o u g h d i r e c t i o na n g l e s ( 0 , ) i n s p h e r i c a l c o o r d i n a t e s . T h e d i r e c t i o n a n g l e s( w h i c h a r e i n g e n e r a l c o m p l e x ) a r e d e f i n e d f r o m t h e r e l a t io n s :

kx = k s i n 0 c o s . ( 3.2 9)k = k s i n 0 s i n . ( 3 . 3 0 )k z = k c o s 0 . ( 3 . 31 )

Homogeneous (Uniform) Plane WaveO n e i m p o r t a n t c la s s o f p la n e w a v e s is th e c l as s o f h o m o g e -n e o u s o r u n i f o r m p l a n e w a v es . A h o m o g e n e o u s p l a n e w av e iso n e f o r w h i c h t h e d i r e c t i o n a n g l e s a re , b y d e f i n i t i o n , r e a l . Ah o m o g e n e o u s p l a n e w a v e en j o y s ce r t ai n s p e c ia l p r o p e r t i e s t h a ta r e n o t t r u e i n g e n e r a l f o r a l l p l a n e w a v e s . F o r s u c h a p l a n ew a v e , t h e w a v e n u m b e r v e c t o r c a n b e w r i t te n a s:

k = k R = R ( k ' - j k " ) ,

w h e r e R i s a r ea l u n i t v e c t o r d e f i n e d f r o m :( 3 . 3 2 )

= k s i n 0 c o s + ~ s i n 0 s in ~ b + ~ c o s 0 . ( 3 . 3 3 )I n t h i s c a s e, th e k v e c t o r i s p r o p o r t i o n a l t o t h e r e a l v e c t o r R ,s o th e r e s u l t o f e q u a t i o n 3 . 27 a p p l i e s . T h e u n i t v e c t o r R t h e ng i ve s t h e d i r e c t i o n o f t im e - a v e r a g e p o w e r f l o w a n d a ls o p o i n t si n t h e d i r e c ti o n o f t h e p h a s e a n d a t t e n u a t i o n v e c t o rs . T h a t i s,a ll t h r e e v e c t o rs p o i n t i n t h e s a m e d i r e c t io n f o r a h o m o g e -n e o u s p l a n e w a v e . T h i s d i r e c t i o n i s , u n a m b i g u o u s l y , t h e d i r e c -t i o n o f p r o p a g a t i o n o f t h e p l a n e w a v e . T h e p l a n e s o f c o n s t a n tp h a s e a r e a ls o t h e n t h e p l a n e s o f c o n s t a n t a m p l i t u d e , b e i n gt h e p l a n e s p e r p e n d i c u l a r t o t h e R v e ct o r . T h a t i s , t h e p l a n ew a v e h a s a u n i f o r m a m p l i t u d e a c r o ss t he p l a n e p e r p e n d i c u l a r


4/12

5 1 6 D a v i d R . J a c k s o nt o t h e d i r e c t i o n o f p r o p a g a t i o n . I f t h e p l a n e w a v e i s n o th o m o g e n e o u s , c o r r e s p o n d i n g t o c o m p l e x d i r e c t i o n a n g l e s ,t h e n t h e p h y s i c a l i n t e r p r e t a t i o n o f th e d i r e c t i o n a n g le s is n o tc lear .

F r o m e q u a t i o n s 3 .1 8 o r 3 . 1 9 , it m a y b e e as il y p r o v e n t h a t t h ef ie ld s o f a h o m o g e n e o u s p l a n e w a v e o b e y t h e r e l a ti o n :

IEJ = In l l HI . (3 . 34)( R e c a ll t h a t a l l p l a n e w a v e s o b e y e q u a t i o n 3 . 2 2 b u t n o t i ng e n e r a l e q u a t i o n 3 . 3 4 ) .

A n y h o m o g e n e o u s p l a n e w a v e c a n b e b r o k e n u p i n t o a s u mo f t w o p l a n e w a v e s , w i t h r e a l e le c t r i c fi e ld v e c t o r s t o w i t h i nm u l t i p l ic a t i v e c o n s t a n t s p o l a r i z e d p e r p e n d i c u l a r t o e a c h o t h e r.T h i s f o ll o w s f r o m a s im p l e r o t a t i o n o f c o o r d i n a t e s ( t h e d i r e c -t i o n o f p r o p a g a t i o n i s t h e n z ' , w i t h e l e c t ri c f ie l d v e c t o r s i n t h ex ' a n d y ' d i r e c t i o n s ) . T h e t w o p l a n e w a v e s h a v e t h e f i e l d s(Ex , F ly ) a n d ( E y , H x ) , w i t h E x / H r = - E y / H x = r I . T h i s de -c o m p o s i t i o n i s u s e d l a t er i n t h e d i s c u s s io n o f p o la r i z a t io n .L o s sl e ss M e d i a : R e l a t i o n B e t w e e n P h a s e a n d A t t e n u a t i o n V e c t or s

n o t h e r i m p o r t a n t s p e c i a l c a s e f o r a p l a n e w a v e c o n c e r n s al o ss l es s m e d i u m , s o t h a t k " = 0 . I f e q u a t i o n 3 . 4 i s s u b s t i t u t e di n t o t h e s e p a r a t i o n e q u a t i o n 3 .1 7 , th e i m a g i n a r y p a r t o f t h ise q u a t i o n i m m e d i a t e l y y i e l d s t h e r e l a t i o n :

f t . a = 0 . ( 3 . 3 5 )H e n c e , f o r a l o ss l es s r e g i o n , t h e p h a s e a n d a t t e n u a t i o n v e c t o r sa r e a l w a y s p e r p e n d i c u l a r . I n s o m e a p p l i c a t i o n s ( e .g . , a F o u r i e rr a n s f o r m s o l u t io n o f r a d i a t io n f r o m a n a p e r tu r e i n a g r o u n dl a n e a t z = 0 o r f r o m a p l a n a r c u r r e n t s o u r c e a t z = 0 [ C l e m -

m o w , 1 9 9 6 ] ) , a p l a n e w a v e p r o p a g a t i n g i n a l o s sl e ss r e g i o n h a s

t h e c h a r a c t e ri s t ic o f t h a t t w o w a v e n u m b e r s ( e. g. , k x a n d k , ) a rer e a l ( c o r r e s p o n d i n g t o t h e t r a n s f o r m v a r i a b l e s ). I n t h i s c a s e , t h et h i r d w a v e n u m b e r k z w i l l b e r e a l i f k2 + k~ < k 2 a nd wil lb e i m a g i n a r y i f k2 + k} > k2 ; tha t i s, a l l t r ans v e rs e wa ven u m b e r s ( k x , k y ) t h a t l ie i n a c i r c le o f r a d i u s k i n t h e w a v en u m b e r p l a n e w i ll b e p r o p a g a t i n g , w h i l e al l w a v e n u m b e r so u t s i d e t h e c i r c l e w i l l b e e v a n e s c e n t . I n t h e f i r st c a se , th e p o w e rf l o w is i n t h e d i r e c t i o n o f th e ( r e al ) k v e c t o r (kx , ky , k z ) , s o p o w e rl e a ve s t h e a p e r t u r e f r o m t h i s p l a n e w a v e . I n t h e s e c o n d c a se , t h ep o w e r f l o w is i n th e d i r e c t i o n o f t h e t ra n s v e r se w a v e n u m b e rv e c t o r ( k x , k y , 0 ) , so n o p o w e r l e av e s t h e a p e r t u r e f o r t h i s p l a n ew a v e c o m p o n e n t . I f t h e m e d i u m is l o ss y , t h e r e is n o s h a r pd i s t i n c t i o n b e t w e e n p r o p a g a t i n g a n d e v a n e s c e n t p l a n e w a v e s .I n t h i s c a s e, al l p l a n e w a v e s c a r r y p o w e r i n t h e d i r e c t i o n o f t h ev e c t o r ( k x , k y , Rekz) .

F i na ll y , i t c a n b e n o t e d t h a t i f a h o m o g e n e o u s p l a n e w a v e i sp r o p a g a t i n g i n a l o s s l e s s r e g i o n , t h e n t h e a t t e n u a t i o n v e c t o r c xm u s t b e z e r o , a n d a ll w a v e n u m b e r c o m p o n e n t s kx , ky , a n d k zare rea l .S u m m a r y o f B a s i c P r o p e r t ie sA s u m m a r y o f th e b a s i c p r o p e r t ie s o f a p l a n e w a v e , d i sc u s s e di n t h e p r e v i o u s s u b s e c t i o n s , i s g i v e n i n T a b l e 3 . 1 .

3 .3 Propagat ion of a H om ogen eousPlane WaveT h e c a se o f a h o m o g e n e o u s p l a n e w a v e is i m p o r t a n t e n o u g ht o w a r r a n t f u r t h e r a t t e n t i o n . F a r a w a y f r o m a t r a n s m i t t i n ga n t e n n a , t h e r a d i a t i o n f i e ld a l w ay s b e h a v e s a s a h o m o g e n e o u sp l a n e w a v e ( i f t h e s p h e r i c al w a v e f r o n t i s a p p r o x i m a t e d a s

3.1 Summ ary of the Basic Properties of a Plane WaveInhomogeneous Homogeneous

Prop erty Lossy Lossless Lossy LosslessDirection anglesRelation betweenphase andattenuation vectorsImpedan ce relations

irection of pow erlow vector

y o f vectors

Complex//. ,* = k', k"E . E = ~ 1 2 H . HNo t necessarilyin fl direction

k.k = k 2

E . H = OE . k = OH . k = O

Complexf l - ~ = OE . E = r I 2 H . H13If k, E0, or H0is proportional toa real vectork .k - k 2

E . H = OE . k = OH . k = O

Real

E . E - - v l 2 H . HIEJ = I< ls l

k . k - k 2k. k* = [k2[E . H = OE . k = OH . k = o

Real

E . E = r l 2 H . HI EI = n l n l

k . k - k2k. k* - k2E . H = OE . k = OH . k = O


5/12

P l a n e W a v e P r o p a g a t i o n a n d R e f le c t i o n 517

a t ion de sc r ibe d by a r e a l un i t ve c to r. B y a su i ta b l eo f c oo r d ina t e a x is , the d i r e c t ion o f p r opa g a t ion m a y

ys be represen ted as a sum of two separa tethe o th e r po l a r i z e d in t he y d ir e c t ion . C ons ide r -

E x = E o e - j k z . (3.36)H y = E o e _ J kz . (3.37)

nnu m b e r k a nd the i n t r i n s i c im pe da nc e q o f t he spa c e

s 3 .12 and 3 .23 , r espec tive ly .

. 3 .1 W a v e l e n g t h , P h a s e V e l o ci ty , a n d G r o u pV e l o c i t y

he w a ve le n g th X i s de fine d a t t he d i s t a nc e r e qu i r e d f o r t helane wave to change p hase b y 2- rr r adians and is g iven by:

2 " / 7h = - - . ( 3 .38 )kThe p ha se ve loc i ty o f t he p l a ne w a ve is de f ine d a s t he ve loc i tya t w h ic h a po in t o f c ons t a n t pha se t r ave l s ( suc h a s the c r e s t o fthe w a ve w he r e t he e l e c t ri c f i eld is t he m a x im um ) . T he pha seve loc i ty i s g iven by:

3 . 3. 2 D e p t h o f P e n e t r a t i o n ( S k i n D e p t h )F o r a l o s sy m e d ium , t he p l a ne w a ve de c a ys a s i t p r opa ga te sbe c a use k" > 0 . The de p th o f pe ne t r a t i on , dp , i s de f ine d as thedis tance required to reduce the f ie ld leve l by a fac tor ofe ~ 2 .71828, so the f ie ld i s 36 .788% of the s ta r t ing va lue ( thepow e r i s re duc e d b y a f a c to r o f e2 or i s a t a leve l of 13 .534% o fthe s t a r t i ng va lue ). The de p th o f pe ne t r a t i on i s g ive n as :

1 - 1d p - k " - I m ( c o x / ~ ) ( 3 . 4 2 )

The pe r m i t t i v i t y i s c om ple x f o r l o ssy m e d ia a nd i s re p r e se n te dby eq ua t io n 3 .10 . Spec ial cases a re of pa r t icu la r in te res t . Forlow - lo ss m e d ia , 8 "


6/12

51 8W he the r a pa r t i c u l a r m a te r i a l f a l l s i n to t he l ow - lo s s o r

igh ly l o s sy c a te go r i es m a y de pe n d on f r e quenc y . F o r e xa m ple ,ons id er typica l seawate r, which has rou ghly gr = 78 ( ignor ingola r iz a t ion losses) a nd ~ = 4 .0 S / re . ( In ac tua l i ty , pola r iza -ion lo s s a l so be c om e s im por t a n t a t m ic r ow a ve a nd h ighe rrequenc ies . ) Table 3 .2 show s the loss tang ent ve rsus f requency.he l os s t a nge n t i s on the o r de r o f un i ty a t m ic r ow a v e fr e que n -ies . For much lower f requenc ies the seawate r i s h ighly lossy;o r m u c h h ighe r f r e que nc ie s, i t is a l ow - lo ss m e d ia . N o te t ha the de p th o f pe ne t r a t i o n c on t inue s t o de c r e ase a s the f r e que nc ys r a is e d , e ve n thou gh the m e d iu m be c om e s m or e o f a l ow - lo s s

a t e r i a l a t h ighe r f r e que nc ie s a c c o r d ing to t he d e f in i t i on u se d( a low - lo s s ta nge n t ) . A t l ow f r e que nc ie s , t he p e ne t r a t i on de p this inc reas ing inverse ly as the square roo t o f f requ encyaccording to equa t ion 3 .48 . At h igh f requenc ies , the pene t ra -ion de p th a pp r oa c h e s a c ons t a n t a c c o r d ing to e qua t ion 3 .46 .

3 .3 .3 S u m m a r y o f H o m o g e n e o u s P la n e W a veProperties

m m a r y o f t he p r ope r t i e s r e l a ti ng to w a ve le ng th , veloc ity ,n d d e p t h o f p e n e t ra t i o n f o r a h o m o g e n e o u s p l a n e w a v e is

. 3 . 4 Polarizat ionhe p r e v ious d i s c us s ion a s sum e d a hom oge ne ous p l a ne w a ve

propaga t ing in the z d i rec t ion , pola r ized wi th the e lec t r ic f ie ldTABLE 3 .2 Loss Tangent and P ene t ra t ion D epth fo r Typical Seawate r

ersus FrequencyFrequency [GHz] Loss tangent (g ' /g ' ) Penetrat ion depth d p [meters]

0.000001 921800 7.9580.00001 92180 2.51650.0001 9218 0.79580.001 921.8 0.25180.01 92.18 0.08000.1 9.218 0.02661.0 0.9218 0.0127

10.0 0.09218 0.01173100.0 0.009218 0.01172

T A B L E 3 . 3 P r o p a g a t io n P r o p e r ti e s o f a H o m o g e n e o u s P l a n e W a veGeneral Highly conducting Low loss Losstess

2~r / 2 2-rr 2~k ~ 2 w V pm o" c o x / ~ c o , f~ gco 2 2 ~ 1 1hase ve loc ity ~7 V ~ x / ~

roup velocity dco 2 ~ 1 1d~ V T a , / ~ , / ~ep th o f pene t ra t ion 1 / l ( ' l / - -

F 2Vco~

D a v i d R . J a c k s o ni n t he x d i r e c t ion . The m os t ge ne r a l po l a r i z a t ion o f a w a vep r opa ga t ing in t he z d i r e c t ion i s one ha v ing bo th x a nd yc om pon e n t s o f t he f ie ld :

E = (~Exo + j IEro ) e - j ( k z ) . (3.49)The f i e ld c om po ne n t s a r e r e p r e se n te d in po la r f o r m a s:

G o = ] G 0 ] d*x. (3.50)G o = I G 0 1 d + y . ( 3 . 5 1 )

The pha se d i f fe r e nc e be tw e e n the tw o c om pon e n t s i s de f ine das :

~b = dO - ~bx. ( 3 . 5 2 )W itho ut any rea l loss of genera li ty , (bx m ay be ch osen as ze ro ,so tha t ~b = ~b . In the t im e do ma in , th e f ie ld com po nen ts a rethe n :

~ x = I Ex 0 1 c o s ( t o t ) (3.53)~ y = G 0 1 c o s ( t o t + 4 ) . ( 3 . 5 4 )

U sing t r i gonom e t r i c i den t i t ie s , e qua t ion 3 . 54 m a y be e xpa n de din to a sum o f s in ( to t) a nd c os ( to t ) func t ions , a nd th e n e qua -t i on 3 .53 m a y be u se d to pu t bo th c os ( t o t) a n d s in ( t o t) i nt e r m s o f 8x ( u s ing s in 2 = 1 - c os 2 ) . A f t e r s im p l i fi c a ti on , t heresult is as follows:

a g 2 x q - B s x g y q - C 8 2 y = D , ( 3 . 5 5 )w he r e t he f o l low ing holds :

E r 2A = ~ x o ' ( 3 . 5 6 )

B = - 2 E r c o s + .Ex0 ( 3 . 5 7 )( 3 . 5 8 )( 3 . 5 9 )

C z l .D = IErol 2 sin 2 ~b.

A f t e r s im p l if i c at i on , t he d i s c r im ina n t o f t h i s qu a d r a t i c c u r ve i s:2

A = B 2 - 4A C = - 4 G 0 s i n 2 dO , ( 3 . 6 0 )I/~x01wh ich is a lways nega t ive . This curve thu s a lways represents ane ll ipse. T he g enera l form of the e l l ipse i s show n in F ig ure 3 .1 .Th e t i l t angle of the e l l ipse is "r, and the a xia l r a t io (AR) isdef ined as the ra t io o f the m ajor axis o f the e l lipse to th e


7/12

P l a n e W a v e P r o p a g a t i o n a n d R e f le c t io n

Y

77 xB2

A1 &A R - B1 B2FIGURE3.1 Ge om etry of the Polarization Ellipse

ino r axis (AR _> 1) . In the t ime dom ain , the e lec t r ic fie ldr ro ta tes wi th the t ip of the vec tor ly ing on the e ll ipse.elliptical polarization ( R H EP ) c o r r e sponds t o

r o t a t i on ( t he t hum b o f t he r i gh t ha nd a li gnsd i r e c t ion o f p r opa ga t ion , a nd the f i nge rs o f t he r i gh td a l ign w i th t he d i r e c t ion o f r o t a t i on in t im e ; t h i s is t he

EEE de f in i t ion , w h ic h i s oppos i t e t o t he u sua l op t i c s c onve n -nds t o r o t a t i on in t he oppo s i t e d i r e ct ion .

A c onve n ie n t w a y to r e p r e se n t t he p o la r i z a t ion s t a t e is w i the t r i c r e l a t ions , t he f o l low ing re su lt s m a y be de r ive d f o r t he

il t angle of the ellipse and the axial ra tio:tan 2~ = tan 2y cos (b, (3 .61)sin 2~ = sin 2y sin qb. (3.62 )

he paramete r { i s r e la ted to the axia l r a t io as := c o t I ( A R ) , - 4 5 < { _ < + 4 5 .

+ f o r L H E P- f o r R H E P

(3.63)

h e phase angle qb i s de f ined in e qua t ion 3 .52 . The pa ram ete rharac te r izes the ra t io of the f ie lds a long the x- and y-axes , and

E yo (3.64)= tan-1 ~ .

-9 0 _< 2{ _< +9 0 . Equa t ion 3 .61 , however , g ives an amb igu-r be c a use a dd ing m u l t i p l e s o f 180 doe s no t c ha nge the

e te rmine the appropr ia te quadrant (1 , 2 , 3 , or 4) the angle 2" rThe fo l lowing spec ia l cases a re impor tant .

(1) Lin ear polarization: qb = 0 o r 180 , or e ith er Exo o rE yo is zero.

51 9TABLE3.4 QuadrantCon taining the A ngle 2-r

cos+ > 0 cos+ < 0cos 2~? > 0 1 4cos 23' < 0 2 3

In th is case , AR = o c ( the e l l ipse degenera tes to a s t ra ightline) .(2 ) Cir cula r polarization: Eyo = Exo and da = +9 0 ( to

w i th in a ny m u l t i p l e o f 180 ).In th is case , the e l l ipse becomes a c i rc le (e i the r RHCP orLH C P ) , a nd A R = 1 .

I t c a n a l so be no te d tha t a w ave o f a r b i t r a r y po la r i z a t ion c a nbe r e p r e se n te d as a sum o f R H C P a nd LH C P w a ves , by no t ingtha t t he a r b i t r a r i l y po la r i z e d w a ve in e qua t ion 3 . 49 c a n bewr i t ten as :

E = ~ [ ~ 2 ( G o + j Ey 0)] + J [ -~ 2 ( G o - J G 0 ) ] , ( 3 .6 5 )w h e r e t h e u n i t - a m p l i t u d e R H C P a n d L H C P w a v e s a r e t h efo l lowing:

1i" = ~ ( ~ - j ~ , ) e - j k z . (3.66)

V ~ + J J ') e - 'k z " (3.67)

3 .4 P l a n e W a v e R e f l e c ti o n a n dT r a n s m i s s i o n

A genera l p lane wave reflect ion and transmission problemc ons is t s o f a n i nc ide n t p l a ne w a ve im p ing ing o n a m u l t i l a ye rs t ruc ture , a s shown in F igure 3 .2 . An impor tant spec ia l case i sthe tw o- r e g ion p r ob le m show n in F igu r e 3 . 3 . The inc ide n tp l a n e w a v e m a y b e e i t h e r h o m o g e n e o u s o r i n h o m o g e n e o u s ,a nd a ny o f t he r e g ions m a y ha ve a n a r b i t r a r y a m o un t o f lo ss .The ve c to r r e p r e se n t ing the d i r e c t ion o f p r opa ga t ion f o r t heinc ide nt wave in F igures 3 .2 and 3 .3 i s a rea l vec to r (as sho wn)i f t he i nc ide n t w a ve i s hom o ge ne o us , b u t t he a na ly si s is va l idfor the genera l case.

The ke y to ob ta in ing a s im p le so lu t ion to suc h r e f l e c t iona nd t r a nsm is s ion p r ob le m s i s t he u se o f a t r a nsm is s ion l i nem o de l o f the l a ye r e d s t r uc tu r e d , w h ic h i s ba se d on TE - T Mde c om pos i t i on o f t he p l a ne w a ves , d i s c us se d ne x t .

3 . 4 . 1 T r a n s v e r s e E l e c t r ic a n d T r a n s v e r se M a g n e t i cD e c o m p o s i t i o nA c c or d ing to a ba s i c e l e c t r om a gne t i c t he o r e m ( H a r r ing ton ,1961 ) t he f ie lds i n a sou r c e - f r e e hom oge n e ous r e g ion c a n be


8/12

20 D a v i d R . J a c k s o n

in c

t r ans

re f

z?

H EH

A Plane Wav e Reflecting fro m a Multilayer Stack ofMaterials. Each layer has a un iform (constant) set of ma ter-parameters. O n th e right side, the equivalent transmission-line

in c : r e fin c

t rans

(

)

T e f

Reflection and Transmission fro m a Single InterfaceDifferent Media. T he transmission-line mo del (trans-

epres ented as the su m of two ty pes o f f ie lds : a f ie ld tha t i sa g n e t i c t o z ( T M z ) a n d a fi el d t h a t is t r a n s v e r s e

l ec t r i c to z (TE~). The TM z f i e ld has Hz = 0 , and the TEzie ld by def in i t ion h as Ez = 0 . A genera l p lan e wave f i e ld ma y

o f a T M z p l a n e w a v e a n d a T E z

n d i c u l a r t o t h e l a y e r s b e c a u s e t h i s a ll o w s f o r t h e T M z a n dE z p l a n e w a v e s i n e a c h r e g i o n t o b e m o d e l e d i n d e p e n d e n t l ys w a v e s o n a t r a n s m i s s i o n l i n e . S t a n d a r d t r a n s m i s s i o n - l i n e

h e o r y m a y t h e n b e c o n v e n i e n t l y u s e d t o s o l v e p l a n e w a v ee f l e c ti o n a n d t r a n s m i s s i o n p r o b l e m s i n a r el a ti v e l y s i m p l ea n n e r , w i t h o u t h a v i n g t o s o lv e t h e e l e c tr o m a g n e t i c b o u n d -

r y - v a l u e p r o b l e m o f m a t c h i n g f ie l ds a t t h e i n t e r fa c e s . T h ethe e l ec t r i c f ie ld " in the p lace o f inc idence ," whereas

T E z p l a c e w a v e i s p o l a r i z e d w i t h t h e e l e ct r ic f i el d " p e r p e n -n c i d e n c e " ( T h e p l a n e o f in c i d e n c e i s

y - z p l a n e .) F o r a h o m o g e n e o u s p l a n e w av e w i t h a ni n th e y - z p l a n e , th e p o l a r i z a t io n s a r e

h o w n i n F i g u r e 3 . 4 .

FIGURE 3.4 Polarizations of Incident TMz and TEz Plane WavesT h e f i el d c o m p o n e n t s m a y b e f o u n d b y u s i n g M a x w e l l' s

e q u a t i o n s t o w r i t e t h e t r a n s v e r s e ( x a n d ) I ) f i e l d c o m p o n e n t si n t e r m s o f th e l o n g i t u d in a l c o m p o n e n t s E z a n d H z . T h en o n z e r o l o n g i t u d i n a l c o m p o n e n t i s a s s u m e d t o b e p r o p o r -t i o n a l t o t h e w a v e f u n c t i o n :

0 = e x p ( - j ( k x x + k y y4 - k z z ) ) . (3 .68)T h e p l u s s i g n i n t h i s e q u a t i o n i s c h o s e n f o r p l a n e w a v e sp r o p a g a t i n g o r d e c a y i n g i n t h e p o s i t i v e z d i r e c t i o n , w h i l e t h en e g a t i v e s ig n i s f o r p r o p a g a t i o n o r d e c a y i n t h e m i n u s zd i r e c t i o n . T h i s r e p r e s e n t a t i o n i s c o n v e n i e n t f o r p l a n e w a v er e f l e c ti o n p r o b l e m s , s in c e t h e c h a r a c t e r is t i c i m p e d a n c e o f at r a n s m i s s i o n l i n e t h a t m o d e l s a p a r t i c u l a r r e g i o n w i l l t h e nh a v e a p o s i t i v e c h a r a c t e r i s t i c i m p e d a n c e f o r b o t h u p w a r d a n dd o w n w a r d p r o p a g a t i n g p l a n e w a v e s , i n a g r e e m e n t w i t h t h eu s u a l t r a n s m i s s i o n l i n e c o n v e n t i o n .

F o r t h e T M z p l a n e w a v e, th e n o r m a l i z e d f i el d c o m p o n e n t sm a y t h e n b e w r i t t e n a s :G = ~ k x k z ' t ' ( x , y , z ) .E y = ~ k y k z ' V ( x , y , z ) .E z = (k 2 - k2z)a tr(x , y , z ) .

H x = o ~ k ? I ~ ( x , y , z ) .G = - o ~ k ? ! q x , y, z ) .

H ~ = O .(3.69)

F o r a T E e , p l a n e w a v e t h e c o r r e s p o n d i n g r e s u l t s a r e t h efo l lowing:E ~ = - o ~ k r ' t q x , y , z ) .G = o ~ k x ' I q x , y , z ) .E z = o .

g ~ = ~ k ~ k z ' t ' ( x , y , z ) .H y = ~ k y k z ' I , ( x , y , z ) .S z = ( k 2 - k ~ ) ~ ( x , y , z ) .

(3.70)

N o t e t h a t a p l a n e w a v e p r o p a g a t i n g i n th e z d i r e c t i o n ( k z = k )h a s b o t h l o n g i t u d i n a l f i e l d c o m p o n e n t s t h a t a r e z e r o . S u c h ap l a n e w a v e i s t r a n s v e r s e e l e c t r i c a n d m a g n e t i c ( T E M ) t o t h e zd i rec t ion .

I t m a y b e s e e n f r o m t h e a b o v e e q u a t i o n s t h a t t h e t r a n s v e r s ef i e lds obey the re l a t ions :

E 7 . IM ) , (3.71)


9/12

P l a n e W a v e P r o p a g a t i o n a n d R e f l e ct i o n 52 1n d

(3.72)Z T E = l ] i s e c 0 i .

U s i n g S n e l l ' s l a w , t h e i m p e d a n c e s t h e n b e c o m e :( 3 . 8 1 )

Z o M = r h 1 - - n l s in e 01. (3.82)Z 0 r ~ = - - . k ~ (3 .73)o2a Z0 E = rli (3.83 )

Cl~ (3.74) 1 / ( n l ~ 2z V e - - k~ 1 - - - s i n 2 01V k n i /3 . 4. 2 T r a n s v e rs e E q u i v a l e n t N e t w o r kF r o m t h e e q u a t i o n s 3 . 7 1 a n d 3 . 7 6 , t h e t r a n s v e r s e ( p e r p e n d i c u -a r to z ) f ie l d c o m p o n e n t s b e h a v e a s v o l t a g e s a n d c u r r e n t s o n ar a n s m i s s io n l in e m o d e l c a l le d t h e t r a n s v e r s e e q u i v a l e n t net-

I n p a r t i c u la r , t h e c o r r e s p o n d e n c e i s g i v e n t h r o u g h t h er e l a ti o n s ( i = T M o r T E ) :

E l = ~ G ( x , y ) V;(z) . (3 .75 )H i t = h t ~ t ( x , y ) I i ( z ) . (3 .76)

I n t h e s e e q u a t i o n s , t h e u n i t v e c t o r s a r e c h o s e n s o t h a t :x h = + L ( 3. 77 )

f o r a p l a n e w a v e p r o p a g a t i n g i n t h e i ~ d i r e c t io n . T h e t r a n s -v e r s e w a v e f u n c t i o n i s w r i t t e n a s:

q 3 t ( x , y ) = e j ( kxx+kyy) , (3.78)h i c h i s a c o m m o n t r a n s v e r s e p h a s e t e r m t h a t m u s t b e t h e

s a m e f o r a l l r e g i o n s ( i f t h e t r a n s v e r s e w a v e n u m b e r s k x or kye r e d i f f e r e n t b e t w e e n t w o r e g i o n s , a m a t c h i n g o f tr a n s v e r s e

f ie l ds a t t h e b o u n d a r y w o u l d n o t b e p o s s i b l e ) . T h e f a c t t h a t t h er a n s v e r s e w a v e n u m b e r s a r e t h e s a m e i n a l l r e g i o n s l e a d s t oh e law of re f lec t ion t h a t s t a t e s t h e d i r e c t i o n a n g l e 0 f o r ae f l e c t e d p l a n e w a v e m u s t e q u a l t h e d i r e c t i o n a n g l e f o r t h enc ident p lane wave . Thi s fac t a l so l eads to Sne l l ' s l aw tha tt a t es the d i re c t ion angles 0 ins ide each o f the reg ion s a rer e l a t e d t o e a c h o t h e r , t h r o u g h t h e f o l l o w i n g :

n i s in 0i = n l s in 0 1 a n d i = 1 , 2 . . . . N , ( 3 .7 9 )n i i s t h e i n d e x o f r e f r a c t io n ( p o s s i b l y c o m p l e x ) o f re g i o n

i , de f ined as n i = k i / k 0 = ~ . I t i s o f t e n c o n v e n i e n t t ox p r e s s t h e c h a r a c t e r i s ti c i m p e d a n c e f o r r e g i o n i f r o m e q u a -

7 3 a n d 3 .7 4 i n t e r m s o f t h e m e d i u m i n t r in s i c i m p e d -n c e r l i as :

z f M = q i c o s Oi, ( 3.8 0 )

T h e a b o v e e x p r e s s i o n s r e m a i n v a l i d f o r l o s s y m e d i a . T h es q u a r e r o o t s a r e c h o s e n s o t h a t t h e r e a l p a r t o f t h e c h a r a c t e r i s -t i c i m p e d a n c e s a r e p o s i t iv e .

T h e f u n c t i o n s V ( z ) a n d I ( z ) b e h a v e a s v o l t a g e a n d c u r r e n to n a t r a n s m i s s i o n l i n e , w i t h c h a r a c t e r i s t i c i m p e d a n c e Z 0 M o rZ 0 E , d e p e n d i n g o n t h e c as e. H e n c e , a n y p l a n e w a v e r e f l e c ti o na n d t r a n s m i s s io n p r o b l e m r e d u c es t o a t r a n s m i s si o n l i n e p r o b -l e m , g i v i n g th e e x a c t s o l u t io n t h a t s a t is f ie s al l b o u n d a r y c o n -d i t i o n s . O n e c o n s e q u e n c e o f th i s is t h a t T M z a n d T E z p l a n ew a v e s d o n o t c o u p l e a t a b o u n d a r y . I f t h e i n c i d e n t p l a n e w a v ei s T M z , f o r e x a m p l e , t h e w a v e s in a l l re g i o n s w i ll r e m a i n T M zp l a n e w a v e s . H e n c e , t h i s s i t u a t i o n c r e a t e s t h e m o t i v a t i o n f o rt h e T M z - T E e d e c o m p o s i t i o n . T h e t r a n s v e r s e e q u i v a l e n t n e t -w o r k f o r t h e m u l t i l a y e r a n d t w o - r e g i o n p r o b l e m s a r e s h o w non the r ig ht s ides of F igures 3 .2 an d 3 .3 , respec t ive ly . Th en e t w o r k m o d e l i s t h e s a m e f o r e i t h e r T M z o r T E z p o l a r i z a t io n ,e x c e p t t h a t t h e c h a r a c t e r is t i c i m p e d a n c e s a r e d i f fe r e n t . I f a ni n c id e n t p l a n e w a v e is a c o m b i n a t i o n o f b o t h T M z a n d T E zw a v e s , t h e t w o p a r t s a r e s o l v e d s e p a r a t e l y a n d t h e n s u m m e d t oge t the to ta l re f l ec ted or t ransmi t t ed f i e ld .3 .4 .3 Spec ia l Case : Tw o-Reg ion Prob lemF o r t h e s i m p l e two-region problem of F igure 3 .3 , the re f l ec teda n d t r a n s m i t t e d v o l t a g e s ( m o d e l i n g t h e t r a n s v e r s e e l e c t r i cf i e lds ) a re represented as :

a n dv r e f ( z ) = F v i n c ( o ) e +J k f z

v t . . . . ( Z ) = T v i n c ( 0 ) e J k ~ Z ) z '

(3 .84)

( 3 . 8 5 )w h e r e t h e r e f l e c t i o n a n d t r a n s m i s s i o n c o e f f ic i e n ts a r e g i v e n b yt h e s t a n d a r d t r a n s m i s s i o n l i n e e q u a t i o n s :

Z ~ 2 ) - - Z g 1 )F - (3.86)a~ 2 ) - }- a~ 1 ) "g = 1 + F. (3.87)

N o t e t h a t a p l u s s i gn i s u s e d i n t h e e x p o n e n t o f e q u a t i o n 3 .8 4t o a c c o u n t f o r u p w a r d p r o p a g a t i o n o f th e r e f l ec t ed w a ve .


10/12

5 2 2 D a v i d R . J a ck s onCritical A ngleI f r e g i o n s 1 a n d 2 a r e lo s s le s s , a n d r e g i o n 1 i s m o r e d e n s e t h a nr e g i o n 2 ( n l > n 2 ) , a n i n c i d e n t a n g l e 0 1 = 0 c w i l l e x i s t f o r w h i c hk~21 = 0 . F r o m S n e l l ' s l a w , t h i s c r i t i c a l a n g l e i s t h e f o l l o w i n g :

0c = s in -1 (n~12 ) (3 .8 8)

W h e n 0 1 > 0 o t h e w a v e n u m b e r k~2) w i l l b e p u r e l y i m a g i n a r y ,o f t h e f o r m k ~ 2 ) = _ _ jf l~ 2 ). I n t h i s c a s e , t h e r e i s n o p o w e r f l o wi n t o t h e s e c o n d r e g i o n b e c a u s e t h e p h a s e v e c t o r i n r e g i o n 2 h a sn o z c o m p o n e n t . I n t h is c a s e, 1 0 0 % o f t h e i n c i d e n t p o w e r i sr e f l e c te d b a c k f r o m t h e i n t e r f a c e . T h e r e a r e , h o w e v e r , f i e l d s s t il lp r e s e n t i n r e g i o n 2 , d e c a y i n g e x p o n e n t i a l l y w i t h d i s t a n c e z . I nr e g i o n 2 , t h e p o w e r f l o w i s i n t h e h o r i z o n t a l d i r e c t i o n o n l y .Brewster AngleF o r l o s s le s s la y e r s , i t is p o s s i b l e t o h a v e 1 0 0 % o f t h e i n c i d e n t

o w e r t r a n s m i t t e d i n t o r e g i o n 2 w i t h n o r e f le c ti o n . T h i s c o r-e s p o n d s t o a m a t c h e d t r a n s m i s s i o n l i n e c i r c u i t , w i t h :2 0 ( 1 ) = 2 0 ( 2 ) ( 3 . 8 9 )

F o r n o n m a g n e t i c l ay e rs , it m a y b e e a si ly s h o w n t h a t t h i sm a t c h i n g e q u a t i o n c a n o n l y b e s a ti s fi e d in t h e T M z c a s e( t h e r e i s a l w a y s a n o n z e r o r e f l e c t i o n c o e f f i c i e n t i n t h e T E ~c a s e, u n le s s t h e t r i v i a l c a se o f i d e n t i c a l m e d i u m i s c o n s i d e r e d ) .T h e a n g l e 0 b , a t w h i c h n o r e f l e c t i o n o c c u r s , i s c a l l e d t h eBrewster a n g l e . F o r t h e T M z c a se , a s i m p l e a l g e br a ic m a n i p u -l a t i o n o f e q u a t i o n 3 . 89 y i e l d s t h e r e s ul t :

ta n 0b = ~ /~ . (3 .9 0)

. 4 . 4 0 r t h o g o n a l i t yh e n t r e a t in g r e f l e ct i o n p r o b l e m s , t h e r e i s m o r e t h a n o n e

w a v e i n a t l e a s t o n e o f t h e r e g i o n s : a w a v e t r a v e l i n g i nd i r e c t i o n ( f o c u s i n g o n t h e z v a r i a t i o n ) a n d a

i o n . O f t e n , c a l c u l a t i n ge r f l o w i n s u c h a r e g i o n i s d e s ir e d . F o r a s in g l e p l a n e w a v e ,

l e x p o w e r d e n s i t y i n t h e z d i r e c t io n ( t h e z c o m p o n e n tf t h e c o m p l e x P o y n t i n g v e c t o r ) i s e q u a l t o t h e c o m p l e x p o w e rl o w i n g o n th e c o r r e s p o n d i n g t r a n s m i s s i o n l in e o f t h e t ra n s -

e q u i v a l e n t n e tw o r k . W h e n b o t h a n i n c id e n t a n d a r e -g o n a l i t y r e s u l t s a r e u s e f u l. T h e s e

s r e l at e d t o p o w e r f l o w i n th e z d ir e c t i o n m a y b eb y d ir e c t c a l cu l a t io n u s i n g t h e f ie ld c o m p o n e n t s i n

c o o r d i n a t es .1. A n o r t h o g o n a l i t y e xi st s b e t w e e n a h o m o g e n e o u s T M z

p l a n e w a v e a n d a h o m o g e n e o u s T E z p l a n e w a v e p r o p a -g a t i n g i n t h e s a m e d i r e c t i o n i n t h e s e n s e t h a t t h ec o m p l e x p o w e r d e n s i t y i n t h e z d i r e c t i o n , S ~, i s t h e

F IG U R E 3 .5 A n In c id e n t Wa v e T ra ve lin g in a S e m i - In f in i t e R e g io no f L o ssl es s G la ss . T h e w a v e im p in g e s o n a n a i r g a p se p a ra t in g th e g l as sreg ion f rom an iden t ica l reg ion be low. Inc iden t , re f lec ted , and t rans-m i t t e d p l a n e w a v e s a re sh o w n .

s u m o f t h e t w o c o m p l e x p o w e r d e n s i ti e s S lz a n d $2z.T h i s i s t r u e f o r a l o s s y o r l o s s le s s m e d i a .

2 . A n o r t h o g o n a l i t y e x is ts b e t w e e n a n i n c i d e n t w a v e a n da r e f l e c t e d w a v e i n a l o s sl e s s m e d i u m ( b o t h a r e e i t h e rT E z o r T M z ) , p ro v i d e d t h e w a v e n u m b e r c o m p o n e n t kzo f t h e t w o w a v e s is re a l. T h e t w o w a v e s a r e o r t h o g o n a li n t h e s e n s e t h a t t h e t i m e - a v e r a g e p o w e r d e n s i t y i n t h ez d i r e c t i o n , R e S z , i s t h e s u m o f t h e t w o t i m e - a v e r a g ep o w e r d e n s i t ie s R e S l z a n d R e S 2 z .

T o il l us t ra t e t h e s e c o n d o r t h o g o n a l i t y p r o p e r t y , c o n s i d e r a ni n c i d e n t p l a n e w a v e t r a v e l i n g i n a g l a s s r e g i o n , i m p i n g i n g o na n a i r g a p t h a t s e p a r a t e s t h e g la s s r e g i o n f r o m a n o t h e r i d e n t i c a lg l a s s r e g i o n , a s s h o w n i n F i g u r e 3 . 5 .

I f t h e i n c i d e n t p l a n e w a v e is b e y o n d t h e c r i ti c a l a n g le , t h ep l a n e w a v e s i n t h e a ir r e g i o n w i ll b e e v a n e s c e n t w i t h a n i m a g i n -a r y v e r t i c a l w a v e n u m b e r k z. E a c h o f t h e t w o p l a n e w a v e s i n t h ea i r r e g i o n ( u p w a r d a n d d o w n w a r d ) t h a t c o n s t i tu t e a s t a n d i n gw a v e f i e l d d o n o t , i n d i v i d u a l l y , h a v e a t i m e - a v e r a g e p o w e r f l o wi n t h e z d i r e c t i o n b e c a u s e k z i s i m a g i n a r y . T h e r e i s a n o v e r a l lp o w e r f l o w i n t h e z d i r e c t i o n i n s i d e t h e a i r re g i o n , h o w e v e r ,b e c a u s e t h e r e i s a t r a n s m i t t e d f i el d i n t h e l o w e r r e g i o n . T h e t o t a lp o w e r f l o w i n th e z d ir e c t i o n i s t h u s n o t t h e s u m o f t h e t w oi n d i v i d u a l p o w e r f l o w s . T h e t w o w a v e s i n t h e a i r r e g i o n a r e n o to r t h o g o n a l , a n d t h e s e c o n d o r t h o g o n a l p r o p e r t y d o e s n o t a p p l ys i n c e t h e w a v e n u m b e r k z i s n o t r e al .

3 .5 E x a m p l e : R e f l e ct i o n o f a n R H C P W a v eA n R H C P p l a n e w a v e a t a f r e q u e n c y o f 1 .0 G H z i s i n c i d e n t o nt h e s u r f a c e o f t h e o c e a n a t a n a n g l e o f 0 = 3 0 . D e t e r m i n e t h ep e r c e n t a g e o f p o w e r t h a t i s re f l ec t e d f r o m t h e o c e a n , a n dc h a r a c t e r i z e t h e p o l a r i z a t i o n o f t h e r e f l e c t e d w a v e ( a x i a l ra t i o ,t i lt a n g l e , a n d h a n d e d n e s s ) . I n a d d i t i o n , d e t e r m i n e t h e f i e ld o ft h e i n h o m o g e n e o u s t r a n s m i t t ed p l a n e w a v e. T h e p a r a m e t e r s o ft h e o c e a n w a t e r a r e a s s u m e d t o b e a~ = 7 8 ( i g n o r i n g p o l a r i z a -t i o n l o s s ) a n d ~ r = 4 S / m .


11/12

P l a n e W a v e P r o p a g a t i o n a n d R e f l e c ti o n5 . 1 S o l u t i o n

e o m e t r y o f th e i n c i d e n t a n d r e f le c t ed w a v e s is s h o w n i n. T h e i n c i d e n t p l a n e w a v e is r e p r e s e n t e d a s :

E inc = Eo[& + i t ( - - j ) ] e - / (~ ,y+k~ 'z ) , (3 .9 1)

a p p l y:1ky = k0 s in 0 = ~ ko . ( 3 . 9 2 )

kzl = ko co s 0 = ~ ko . (3 .9 3)2/~ = ~ c o s 0 - ~ s in 0 . (3 . 9 4 )

E ref = E0 [A~ q- B i b le - j G y - k z, z) , ( 3 . 9 5 )

i , = - j , c o s 0 - ~: s in O. ( 3 . 9 6 )T h e x c o m p o n e n t o f t h e i n c i d e n t a n d r e f le c t ed w a v e s c o r r e s-p o n d s t o T E z w a v e s , w h e r e a s t h e u a n d v c o m p o n e n t s c o r r e s-p o n d t o T M z w a v e s . ( T h e u a n d v d i r e c ti o n s s u b s t i t u t e f o rt h e y d i r e c t i o n i n t h e p r e v i o u s d i s c u s s i o n o n p o l a r i z a t i o n . )T h e t r a n s v er s e e q u i v a l e n t n e t w o r k i s s h o w n i n F i g u r e 3 .3 .F r o m e q u a t i o n s 3 . 8 2 a n d 3 . 8 3 , t h e i m p e d a n c e s a r e Z ~ M= 3 2 6 . 2 5 8 1 2 , ZT1E = 4 3 5 . 0 1 1 1 2 , Z [ v = 3 4 .0 5 1 + j ( 1 3 . 2 6 9 ) ,a n d Z f E = 3 4 . 0 8 9 + j ( 1 3 . 3 4 6 ) f L T h e r e f l e c t i o n c o e f f i ci e n t s a r et h e n F T ~ = - 0 . 8 0 8 5 3 5 + j (0 .0 6 66 0 0) a n d F r Z = - 0 . 8 5 3 1 6 1+ j ( 0 . 0 5 2 7 2 4 ) .

T h e c o e f f i c i e n t A i n e q u a t i o n 3 .9 5 is d e t e r m i n e d d i r e c t l yfrom:A = F TE = - 0 . 8 5 3 1 6 1 + j ( 0 . 0 5 2 7 2 4 ) . ( 3 . 97 )

/ N / x ,Id I14 n o : p

~ y

Z

I G U R E 3. 6 G e o m e t r y th a t D e f in e s t h e C o o r d i n a t e S y st e m f o r t h ep le o f a P l a n e Wa v e R ef le c tin g f ro m th e S u r fa c e o f th e O c e a n

5 2 3T o d e t e r m i n e t h e c o e f f i c i e n t B , r e c a l l t h a t t h e v o l t a g e s i n t h et r a n s v e rs e e q u i v a l e n t n e t w o r k m o d e l o n l y t h e t r a n sv e r s e ( h o r i -z o n t a l) c o m p o n e n t o f t h e e l ec t ri c fi e ld ( t h e y c o m p o n e n t i nt h e T M c a s e ) . H e n c e , t h e f o l l o w i n g e q u a t i o n r e s u l t s :

- B E 0 c o s 0 r ef ~ - F T M E o ( - - j ) c o s 0 i n c. ( 3 . 9 8 )S i n c e 0 inc = 0 r ef = 0 , B i s d e t e r m i n e d d i r e c t l y a s :

B = F T M ( j ) = - - 0 . 0 6 6 6 0 0 - - j ( 0 . 8 0 8 5 3 5 ) . ( 3 . 9 9 )T h e p e r c e n t p o w e r r e f l e c t e d i s c a l c u l a t e d f r o m :

( i [ IAI2 IBcs0 .2Dref L z? ~ + z H J % = i 0 0 [#+l_jcos012]zrM, ( 3 . 1 0 0 )

w h i c h y i e l d s :P ~ f = 6 9 . 4 4 1 . ( 3 . 1 0 1 )

H e n c e , 3 0 . 5 5 9 % o f t h e i n c i d e n t p o w e r i s t r a n s m i t t e d i n t o t h eo c e a n .

T h e p h a s e a n g l e b e t w e e n t h e v a n d x c o m p o n e n t s o f t h er e f l e c t e d f i e l d i s a s w r i t t e n h e r e :

d ) = IB - L A = 8 8 . 8 2 7 . (3 . 1 0 2 )H e n c e , t h e r e f l e c t e d w a v e i s a l e f t - h a n d e d e l l ip t i c al l y p o l a r i z e dw a v e .T h e p a r a m e t e r ~ / is d e t e r m i n e d f r o m t h e r a ti o o f t h e m a g n i -t u d e s o f th e v a n d x c o m p o n e n t s o f th e r e f l e c t e d fi e ld as :

y = t a n - 1 ( [ B I ' ~ = 4 3 . 5 0 4 . ( 3 . 1 0 3 )\ I A I JF r o m t h e f o r m u l a s i n t h e p o l a r iz a t i o n s e c t i o n , S u b s e c t i o n 3 . 34 ,t h e t i l t a n g l e "r o f t h e p o l a r i z a t i o n e l li p s e f r o m t h e x - a x i s( t o w a r d t h e p o s i t i v e v - a x is ) i s t h e n :

r = 1 0 . 6 9 0 , (3 . 1 0 4 )a n d t h e p o l a r i z a t i o n p a r a m e t e r ~ i s t h e f o l lo w i n g :

= 4 3 . 3 9 3 . ( 3 . 1 0 5 )T h e a x i a l r a ti o o f t h e r e f l e c t e d w a v e i s th e n :

A R = + c o t ~ = 1 .0 5 8. ( 3 . 1 0 6 )T h e r e f l e c t e d w a v e i s n e a r l y C P b e c a u s e t h e o c e a n i s h i g h l yr e f l e c ti n g ( i f t h e o c e a n w e r e r e p l a c e d b y a p e r f e c t c o n d u c t o r ,t h e r e f le c t ed w a v e w o u l d b e L H C P ) .


12/12

2 4T h e t r a n s m i t t e d f i e l d i n t h e o c e a n i s r e p r e s e n t e d a s :

E rans -~- Eo [~C "[- ~ D + $ E ] e - j (k ry - k z2 3 ) , ( 3 . 1 0 7 )kz2 i s t h e v e rt i ca l w a v e n u m b e r i n t h e o c e a n f o u n d f r o m

kz2 = k 2 c o s 0 2 = k 0 n 2 1 - s i n 2 0 ( 3 .1 0 8 )= k 0 ( 9 . 5 8 2 5 2 7 - j 3 . 7 5 1 6 4 2 ) .

C = T wE = 1 + F TE = 0 . 1 4 6 8 3 9 + j ( 0 . 0 5 2 7 2 4 ) , ( 3 . 1 0 9 )n d

D = T T M ( - j c o s 0 ) = ( 1 AV FT M )( - j c o s 0 )= 0 . 0 5 7 6 7 8 - j ( 0 . 1 6 5 8 1 3 ) . ( 3 . 1 1 0 )p o n e n t o f t h e t r a n sm i t t e d f ie ld c a n b e d e t e r m i n e d

1E = - ~ - ( k v D ) . ( 3 . 1 1 1 )/ ~ z 2 J

D a v i d R . J a c k so nH e n c e , t h e f o l l o w i n g r e s u l t i s o b t a i n e d :

E = - 0 . 0 0 5 5 4 6 6 + j ( 0 . 0 0 6 4 8 0 3 0 ) . ( 3 .1 1 2 )F r o m t h e w a v e n u m b e r s k r a n d kz2, t h e d i r e c t i o n a n g l e s o f th et r a n s m i t te d w a v e m a y b e d e t e rm i n e d f r o m e q u a t io n s 3 . 2 9t h r o u g h 3 . 3 1 . T h e r e s u l t s a r e a s f o l l o w s :

= 90 , (3 . 113)0 2 = 0 . 0 4 5 2 2 7 + j ( 0 . 0 1 7 6 7 9 ) r a d i a n s . ( 3 .1 1 4 )

T h e t r a n s m i t t e d p l a n e w a v e i s i n h o m o g e n e o u s b e c a u s e t h ea n g l e 0 2 i s c o m p l e x .

R e f e r e n c e sClemm ow, P. c. (1996). The plane wave spectrum representation of

electromag netic fields. New York: IEEE Press.Collin, R. E. (1991). Field theory ofguided waves. (2d ed. ) . New York:IEEE Press.

Kraus, J. D. (1988). Antennas. New York: McGraw-Hil l.Harrington, R. F. (1961). Time-harmonic electromagnetic fields. N ew

York: McGraw-Hill.Hayt, W. H. (1989). Engineering electromagnetics. (5 h ed.). Ne w York:

McGraw-Hill.

Documents

Plane Wave Propagation and Reflection