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Problem Set Solutions 4

The document contains exercises related to electromagnetic fields, specifically focusing on TE and TM polarizations and their respective power and phase characteristics. It includes various mathematical expressions and graphs illustrating the relationships between angles and polarization states. The exercises aim to deepen understanding of wave behavior in different media with varying refractive indices.

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ANDREW PERDIKIS
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0% found this document useful (0 votes)
8 views22 pages

Problem Set Solutions 4

The document contains exercises related to electromagnetic fields, specifically focusing on TE and TM polarizations and their respective power and phase characteristics. It includes various mathematical expressions and graphs illustrating the relationships between angles and polarization states. The exercises aim to deepen understanding of wave behavior in different media with varying refractive indices.

Uploaded by

ANDREW PERDIKIS
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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ΗΛΕΚΤΡΟΜΑΓΝΗΤΙΚΑ ΠΕΔΙΑ Α ΣΕΙΡΑ ΑΣΚΗΣΕΩΝ Νο.

4
ΑΣΚΗΣΗ 1:
_____
AvdKAo<;;r-, '" 610 ~.Ao.E.J"
---=---~---.:.-
k<"J~a'lwv
En I I"TE&wv _ _l 6c J:rr; i1E~1"I L\ IQXw-, [

pll;\:Ii<n Enl~clV~li:::t. f-\E"(QSU 2. AIf'\~\~u'te(L(Wv - n~pirrTw<;t1 n~Aw-1

6n s II n TM.

f-' QOVY,\,,(:( U Q nf G,'a ~ (vJ i


~ ~ i

.J f-lr- "H.f:­ ~o npa~nl'nTav) Tal

) ~ ==­ k:2 ( S' Y1 8 2 ~x + co S 8 2 II )


1<2 = W J~2.€2 ' I

rrOll ) 0. v-ri 6'LOI xov.s If 06 \ &i"CE S 'loO\J yt,AEKTp IK'OJ TIi:sIQV,


I
~ ~

E.
~ A ~

k·L- x H'L - WEE


1
. ..:/
(., L---
1<'1 ( k""t x r.)
-\
-
­
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..z:.1
(k.)(~')
L l

I
0[100

/'­ A
~ Ly L~
c::. _
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o CC)(Jl

I
I1Le-jkr. 0

~-------~-----------L.-
---+---------'--------I------,----------i-­
k.:-r -j
~ -!>

r--)

I
......:. ~ ......:.

~J.~?)
Kr ')<. ~r - -WE, Er
/' A A
~ Lx l)' Li!
Er- = -z 1 Sintly- 0 -CO) 9,..... ­ + Hr sm8,- e-
0 \-( (e_jk..r 0

Hiy (Z=o) ­ H2. )I (2=0)

(1)( ('2 ~o) - E 2X Cr == 0)

VOl. <!pa. ~ evv w ~ ~ s As:


- j k2 <;; \ Ii ~2. )(
H{ e

er - 8!. ( ((W v,'Q\ C(Vo.,-,<r\o.6ns -= (j'wv1o. np66 nn.,JGn s ')

k1 sln()1 = k2 slnS2 A kp, ~\'(\81 =­ J'-2}-!'2. 5'\Y\~2.


( v~\-A0~ Lt:\1j S,n,JJ)

---------------------_-----.JL
lA pO(

Av Op,'60U I'-' E. 't'OU J 6<.JVr~A f GTE'S

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, I

(lpan(jo v f-1€.lI E S'


(/)
1-\ Ht '" HL
~
w
w
x:
(/)
o
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o
N

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N

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z, CO s 8. - ~2ccs32..
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t~ =
I

(3 ~ G '" TQ 1\ c\ Lu q:J(\.~0(
- I
l aCt:.

2: 1 cos8 1 -Z:2 cO$8 2


'II -E'r
E, L
:::
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ill l-\ l'
l-J r :0-
HI'
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=:
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til :=: -Et ~


l2. Hi; -
- C2 t = 2 :z.2.. CosB 1
EC 21 1--\ l Z( #
z:! cos 8 1 + Z-2 cos 82

1-\ v x p n (; I \-t C \I 0 \ ~ 0 a \j ~t cS s; \;( t E5

,
kD. U~\ua.) '""C6"t.f. ll:=- ZO/n 1 OnoT~
I

n 2. CO ) 8 1 - n f C-o S 82.
t'J'lccs6, +n 1 (OS 82
2 n I cos8,
I
~t----------------------------------+---

S1 -21 ~
EI ~
x 1-\ 1
*" -- i (t 1 X l )( A 4- Ef r ~l ) )(
( ~~~ t y )

"j{ I'
-- " E 1>< ~1>,'* "l~
:2
+ --1
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E:,~ \-\ 1)' (-Lx)

A
--
A

51 ~ L'2; -l­
S~X Lx

; Z,ccs8 , [leliI2--l~rI'J + ~ 2j 1M {l\\.-\'H:C<:>\31e~J(k~-k!r·'j


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(.r'
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~
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[
"
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~

kl . r + H~
r e
Tj
~

kr " r J i
i

2 "2­
-- I
-
2 [ z: IH
1 l' \ Sin 8:L + 2)Hrl sIn81 +
~ ....> -3

+j(k,'-k()·r
+ 2:, Hi l-\( e
~
]
=

~
2
7
1::: I '-.>
~()"'8

(1 1-1,' \2_1.1-1(/'2.)
S1X Lx
5:1"f\V nEplox~ 2­
~ l ~ ~,I(
S2:= 2. E 2 x H2. ­
t t z~
~
H 2)'
A
,X

~ [~2Ht co'El2 e-J~;= ] [H: e +)k;,"]

S2X - - -2
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l-\1)1
1 [
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2
SIn 82.

<N > - ~ Zz \ H.t/2. cOS 62­


~ ~

2 Re t S2 J 2

-= 0

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2, co ~ 0'1

~yI Zz.. c(Jj~t.


I ~
'l
+ Z:, c~} 0 I
I H~
1..\ L
}

1 - I r-'l J
2
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G-r.n
<

t x: cJ\,) v l''0...po'.J ~ \ ~\:, ~ ~

-
n1 = 1 n2 = 2.2 n1 = 1 n2 = 2.2
1 Phase of rTE
100
|rTE|

0.5 0

−100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
n1 = 1 n2 = 2.2 n1 = 1 n2 = 2.2
1
Phase of rTM

100
|rTM|

0.5 0

−100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
n1 = 1 n2 = 2.2 n1 = 1 n2 = 2.2
0.6 Phase of tTE
100
0.4
|tTE|

0
0.2
−100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
n1 = 1 n2 = 2.2 n1 = 1 n2 = 2.2
0.6
Phase of tTM

100
0.4
|tTM|

0
0.2
−100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
TE Polarization, n1 = 1, n2 = 2.2
1
R and T Power

R
T
0.5

0
0 10 20 30 40 50 60 70 80 90
Angle θ1 (deg)
TM Polarization, n1 = 1, n2 = 2.2
1
R and T Power

R
T
0.5

0
0 10 20 30 40 50 60 70 80 90
Angle θ1 (deg)
n1 = 2.2 n2 = 1 n1 = 2.2 n2 = 1
1 Phase of rTE
100
|rTE|

0.5 0

−100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
n1 = 2.2 n2 = 1 n1 = 2.2 n2 = 1
1
Phase of rTM

100
|rTM|

0.5 0

−100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
n1 = 2.2 n2 = 1 n1 = 2.2 n2 = 1
Phase of tTE
1.5 100
|tTE|

1 0
0.5 −100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
n1 = 2.2 n2 = 1 n1 = 2.2 n2 = 1

4
Phase of tTM

100
|tTM|

2 0

−100
0
0 20 40 60 80 0 20 40 60 80
Angle θ1 (deg) Angle θ1 (deg)
TE Polarization, n1 = 2.2, n2 = 1
1
R and T Power

R
T
0.5

0
0 10 20 30 40 50 60 70 80 90
Angle θ1 (deg)
TM Polarization, n1 = 2.2, n2 = 1
1
R and T Power

R
T
0.5

0
0 10 20 30 40 50 60 70 80 90
Angle θ1 (deg)
ΑΣΚΗΣΗ 2:

(0<)
En! II [00', l<U~Cl:tCJ,:
--"
E·L ~

k·L -= k0 ( Co s 9 ~x + SIn 9 ~~ )

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/' -J k o (X c,.¢) S +)J ~II'\.~)
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Kc (-cosS ~x +- sinS L.::J)
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Ev.C) l z e
~ ( co59 lX - Sl\"\S L!:})
c Lr e -J k';) ( >< co> 9 - ~ J' "-.9 )
do

~ ~ -" ~

(Ei.+Er+Ecl+E",)[ =0 >
~==o

+jkoxcos8 -J'koxcose
Eo e + Era e +

'==1

- - - ~----- _.~-----~--------.
E Jo = - Eo

EI,r<::l =: - l? 0-.::1 -::: - (- Eo) ~ Eo


I

Eno \J or VW) "'t"'~ U\ If ( \ 'L8 t1... ~ f-J tX LCX JvOJ...


~ A -j k:l (- x co S'C(j - :1'" \'\. 3 )
bl = Ie Eo e
-J ko ( xC-(J,S +:j rlh..3 )
e
_j k... ( x Cc S ~ - ~ !> \ n. s )
e
~ k.e (-y Ca ) Sl +:J" i "" ~ )
e

(~ )

Jkoxms~ jko~s\\'\~ -jk..~S'Y'\SJ


e [ ~ -e +
e
-~k)(~~~
[ e
-j k.:.C:j .s1Y'-~
- e
J k.. j s, "-~ J7'( =
.J

1\
: Ie 2-E o j sin (\<o~ slr\9) (2J) SIy"\ ( ~XCO$-3)

- c- 4 E () ) 'Ie Sin ( k<> x co.s 8) s ~ Y) \ k ~ S"1:J)


0

---'------- - - - - - - - - - - - - - ----'---­
I

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rn (JJo )
IT
x=
k,. cos~
::.
2
),,'"
r1 C<::\)'I)
- .
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Qn .err­ Ao e(~O )
~ ==- - ~

k'tl slntJ 2,. r,""'9 2 10'\"\8

,
r'()
)
Q VI a.

,
x =- a ~ vno...pxolJv

J / ,
1"'I()l~ 67~ Hlll'\[S"" x::::o, 1"\ ~=o.

_ _ _I_ d E~ ~:x
jw~o dj

+ -!- ~
jw r
o 21'1(
I\­
Hx ___1_ (~4Eo) s,·n(k:>xc().s~) kcslnS co~(ko~s,,",s)
jW)J<>

....L- (-4E o ) k~c().s~ <:.0) (k",x c.o)~) >I'f) (leo.:! SIn.&-)


j\.J.Jr~

1<::1 ~ y, ~):o 12 (~: ) CO,:J So in (koj sinS) ec, (wt+ "'/2)


:::: -I~ It-co cC),)S S;n (k",,'jSI~~) :s-i'"'(~•.)"t)
x...
------------- ---.­
--'-----~------'--------~--~-~---------+---I
l
r---­

~ -k k
'
f (x=' a) (B(x::-o~) +- B~~-)) ­ j )<

i
~ 0 H'y ( oL~
-'l
-= 2 l j ( :::J It-) x :J) -t ) Ix ::0

f=: ~
-
2
A
(-Ie)
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ro
CD 59 si Y") (k)~ 51 Y\ 8 ) SIl'\ (w+) x r 4-
<>
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:bo
I'
(-I j ) CO)S

S\(\(~~s!n~) SIy\(U+)
2
-1)( " A \6 Eo
_p_
2 .,. ~'f>2
Cos
2.
e sir?- (ko~ S'I'Y\S) s; y\2.(VJt)

...... 4 Eo?
<f> -- "­
-2 co S ~ 0 S'\ ",? (\.<... j s·, '>\ ~ )
-1)<- ~o 2.
0

---.L " _
ΑΣΚΗΣΗ 3:

/
/

,co/ ) d 5\) v~t' n n au (A(,\,..\ L~"()( I


pcxo""
2 ~66w "fou OtcS,'00 1"00

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d 1f /\ II T2 00
"1 2 y~
("
L,? x Aly \
)
f1112;:;:- LX
d z e 2nd 21\ d
I
0yOPP<:loO( PC,U I-' c:.'lCi .I ,
(XI"UPPO(1C!; f~-v VCl."to( (lI Jz<'o)
(rJz ><::»

~ -t f~~2
-'
IM7
~ e­
(I flYl \

61:0\/

claw C C
I
2 '/1'0 61.J TC)v

- J 2. clO:2,
n
.....-I,i

2 nco 1"'11
(1r\1 Lx ) =)

....l

d r2-
~

::J~,12 /',
+e'2-. L:x
d.Q2. 2n Eo d
E-T l' V\ I~o, f{:lopn ex,
((jI;:.J~<a)

- ~

+e2­
rE'-l.

?'i ;:)2­

-'
1 t I 2 P::> A
L A2.. l+
frn~.:.. ::c. -
!y
. !
- 411.10 ­
2T1d ZIT im rn
2- :t.
2.10-+
A SLs -?- --r
--..J
2.1t) Nt
- 2. ~o
Z
-
rr\ 2- rn YY1..

.» 2.
.1
2
)
~Q
~n~<J. 0'd­

~2 ;;;::. )Jaf".-c
_. I?/C2.

J ~
1.
. lA ­
C 3,(oS'mls

3. , -/?
7
ΑΣΚΗΣΗ 4:

')(
,
TEA<:-IO$ O(Su.l~O.s
....a.
~ S'" 12:;1.=0
kr J_ !
-'"
Hl.':. 0
I
I i65
~~

I
I

C] 1'": I
Q)
.....
kl
S(2­ I

At
S ...

(0..) TE nO~Ul6Y) : ( H).El<rpIIC'O n E-~icd


-' ,.. 'k
-J 0
(Zc..ose+)(sil"le)
Et = 1)1 Ei e
(_ 'leaSe + ""sine)
'"1)1 E re -Jko
->

Er ::::
I
I

I
I
l,~ ..
===
E' (
--!:-
",. A) ~.J k' ·r
-cos'1)Ix-+sII')S'l E:l- I

o Zo I
E(
-J ko [lc.os~+)(sinS J
0
I
"-) e
..,.
I
e-f_~LC',r
A
'"t;, tc
'k.-,r­
Sln~J.1
_j
fr [/'­ ",
51 nt:J 0 -c.os'O e - - l)( o:)s e -t I-Z

o L 0
Zo I
F... ['LxcoSB+'lcsineJ e
_j 1.<0 [-i! cooS S -+ x S 1('1 ~ J I
zo
~U\jSrl\<~ S (2:: 0):

o = E'L.

('Ln ~ r t ) )( ( ~l - H~) I:. K ~')


z::co

_ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _-L-_ __
+-------------"--------------''-----___~___ '''",,_+_l-­
A""
_H,\x [ (Iz""x) ~
E' J 'ltv<
(-cese) + 2E co.s0 ei S.I"'~
lo Zo::.
J
I
I

I ,
OIiCu 0. VTIE"O IX00V

l, Kou 'LD- o.t4c!().. CLlf'CI(,,"tCIXC0V € I tO\)5. \f>uE:,,€h''t.,E:S.


I ~
fm = i LKe JUll + K"e-jcO+ J x ~ ~ (B~~j",," +tB")" e-J"" ;­

.1 "B-e-tJ'''')+ +(6-( e-jw+J


t
_Ke [-->~oj( -;w+
jw+ +ke 1x [C8++S-) 2­
e j l.>l-+ + Cl3++B-Y e-jw+
f
J=
8

I
o XPOVIt<os fJE,"os opos 8vO-l:

:::: ~ RefkY­ ~ [tl+f1oHt}j ~


K= ~ y .§i 2. c.os e e -j k.. x s i I') e
L..
·T---------------------~--------------------h-----------------------!---

==

=
,
H 6\JvI" T0)" Q 0' lOt -r-.,,, TE

o t-' w> = 0 J
~ ~~vY\"Cw ~ n \ '€ o,8ov"t:~
I .
t~ ) Ol Vi )cu"tpo '~I.5 Cl.rt~ -US E.X€-E; fl S'

~
[~ ~~ ,~2" ]
Ie - Eo E (C-'''') - Z\C\ IY) 1<Q.l

T"", :=. po
~
[-ge C~'II'I)
-1
-l"I I~
~ r~"
....
l"
J

\
2 4
f [
..-:> --
E.e.e j2.w.- + ... -
E,f..e:"~ ~ ~ ........
-j2A..l+- + ~.E"'-+-E".E J" I",
1
= [0 {-i [Re.(£(£.f",)eJ2O>'l + Re. (E(E·.~~\)J
I
2
1.
l.
[RefE.ee.i2...l~3
.
+RetE.{;Jt} J'LI1'} r--J

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~'1.

<-rc..)
~
= ~ '" "2 R~ [
-- ~
E (E *.'I n ) J \ ~ 2
II')
CO [ A 1
Re fL t:~ (E-"It . I",) (£ r~
- '2 '.,
~G
,
O)JO'-OC.

<1m> := ~o [ Re ~ H(H#-·i'n) -I~f~ 1 ]

Ec.pt>.p,....';'jcV1:0S 'ttl Y\/~ ill u.. u! ~ro Cp~O~fO rTc..poq;l>'\.lE\li(\{~o

CiAE.-UpdJ tt-'~aS()J A~ UCA f"l1.::bcO u ..) Ac iXcufAE;

<Ie" )
..lo.

1(Te, ) ds ~ S<Tel> dS+ + J <T e, '> dS_ + S<ft!, > dSI


s+ S - Sl

-= J[~ I<e ~ ->


E1 (~~ A
£1 ., >,-) J - 1£,1 2
7 ",ll( J d S ... +

s+

Re i E, ,. ) ...... '2.
~ (~~ IE \ I ('" } dS
J
1(;'0
E, '(-he) - - -1)( - +
2 2
$_

R~ ~ £,
~ J -
IE,\ 2­

S
A
Eo -"t.
(£, .,~) . - - '2' dS 1\
A


2.
S,

1< Te >dS
S
:= 0

I
niEU)s. (COW) f(.D-t ~"n" nlp,nTW6Y' r-ns SUVOq.AflS

Lotent'Z) .

s+ ...a,,,,

S~ [Re ~ ~I ( H~ . (-I)d ) 1- Jd S_
IHi I(- ~) +
s_
I";' (Ii ] rS +
~ 2

S
S,
f'; [Re Fi, (H,* {-ill} - 1 S<f;-:.>dS,
0
.... 2 -'" 2.
..... 1)(.
Re { HI HI')J
(~'It A
1)(
_ \\.4,1
2-
,A
Ix
~
I~II
'2
- Hoot" ""Ix
-"Z
R~ l -
HI C~C*-(-I)( ))J H" "
-2 . - 1.
11-1, I (-I'x)
- - '"1>< 1-"
HI \2 + I-
2..
Ix

~ <f'>
<f);--=IZ-H 1
A )..10 I~ \2
AS 4

pL -

Eo 2P~ Zo :::
c

= 6' 6 ..,T 1 . 10- 6 J. ::: b,f.11· '0- 6 N/


m
2

__ N· m .. N/ 2­
- rill m

l<t>1
6 .fiU\-, NI.....2.

~------ ______________ ~~ ______ ~~e

n
o 2.

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