electromagnetic 9

8/11/2019 Electromagnetic 9

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C H A T E R

WaveReflection

|

-r

an(l I ransmlsslon

EM Waves

at Boundaries

9-1 Wave

Reflection

and Transmission

t Normal

Incidence

9-2

Snell'sLaws

9-3 Fiber

Optics

9-4

Wave Reflection

and

Tiansmission t

Oblique

Incidence

9-5

ReflectivityandTiansmissivity

@

* E

l D l

r:LJ

- - E

- a

v

\ lI-'

o

Minor

o)

Mirr

(a)

* -

4 . _ i



EM

WnvTs

T

BOUNDARIEs

The

sketch

n Fig.

9-l

depicts

he

propagation

ath

hat

a signal

ravels

between

a shipboard

ransmitter

and

a

re-

celveron

a

submerged

ubmarine.

et

us

use

his

com-

munication

ystem

o

examine

hewave-related

rocesses

fiat take place

along

the

signal

path.

Starting

with

the

transmit3erdenoted

Tx

for

short

in

Fig.

9_l),

the

sir_

nal travels

along

a transmission

ine

to tf,e

antennu.

Tli.

relationship

etween

he

transmitter

lgenerator;

utpui

power,

P,,.and

hepower

upplied

o

the

antenna

s gov_

emed

y

the

ransmission_line

quations

iven

n

Ctap_

ter8. If the ransmissionine s approximatelyossless

and f

it

is

properly

matched

o

the

ransmittei

ntenna,

then

all of

Pr

s

delivered

o

the

antenna.

he

next

wave_

related rocess

s

that

ofradiation;

hat

s,

converting

he

guided

waveprovided

o

the

antenna

y

the

ransmis"slon

line

nto

a spherical

wave

adiated

ut*_O

into

.pu"".

The

adiation

process

s

the

subject

of

Chapter

O.e.o,

polnt

, denoting

he

ocation

fthe

shipboard

ntenna,

o

point

2, denoting

he point

of incidenci

of

the

wave

onto

thewater's

urface,

he

signal

s

govemed

ytheequadons

cnafactenzlng

ave

propagation

n

a lossless

medium,

which.we

overedn Chapter . As the*aue impinge.

li111Le

air-wler

boundary,

parr

of

it

gets

eflectedly

the

surtace

nd

another

art

gets

ransmitted

cross

he

Doundary

nto

ie

water

medium.

he

ransmitted

art n_

dergoes

efraction.

wherein

he

direction

of *uu"

t.uu.t

moves

loser

oward

he

vertical.

he

eflection

nd

rans_

mission

rocesses

re

reated

n this

chapter.

Wave

ravel

frompoint

3, representing

point ust

below

he

water

surtace,

o point

4.

denodng

he

ocation

of

the

subma_

nne

antenna,

s

subject

o he

aws

f wave

ropagation

n

a

lossy

medium,

which

also

were

reatedin

-Cliapter

7.

The

final

step

nvolves

ntercepting

he

wave

ncident

upon hereceiverantenna ndconverting

ts

power

nto

Transmitter

antenna

\ r

Figure

9-1:

Signal path

between

a shipboard

ransmitter

(Tx)

and

a

submarine

eceiver

Rx).

a received

power,

p,.",

for

delivery

via

a

transmission

Iine

to

the receiver.The receivingpropertiesof anten_

nas

are

covered

n

Chapter

10.

In

summary,

hen,

ea&

wave-related

spect

fthe

transmission

rocess

epicteJ

in.Fig.

9-1,

starting

with

the

nansminei

and

ending

up

with

the

receiver,

s

treated

n

some

section

or

chaptlr

in

this

book.

This

chapter

egins

with

examinations

f

the

eflection

and

tansmission

ropenies

fplane

aves

hen

ncident

upon tanar

oundaries

long

he

normal_incidence

irec_

non.

I hen-

Snell's

aws

of

reflection

nd

efraction

re

applied

o

he

general

ase

foblique

ncidence

v

a

plane

wave.

3i l



312

9-1

Wave ellectionnd

tansmission

at {ormal

ncidence

Weknow

romChapter8

hatwhen guided

wave raveling

along transmission

ine

encounters

n

mpedanceiscon-

tinuity, uch s hat hownnFig.9-2(a) t heboundarybe-

tween

wo lineswith

differentcharacteristic

mp€dances,

the ncident

wave s

partly

reflected

ack oward hesource

and

partly

Fansmitted

crossheboundary

nto hesecond

line. A

similar

process pplies o

a uniform

plane

wave

propagatingn an

nbounded

mediumwhentencounters

a boundary.n fact,

hesituation

epicted

n Fig.

9-2(b)

s

exacdy

analogouso

the transmissionJine

onfiguration

ofFig. 9-2(a).

heboundary

onditions

oveming

he e-

, - n

(a)

Boundary

between

ransmission ines

z = 0

(b)

Boundary between

different media

Figure

9-2: Discontinuity between

wo different

trans-

mission ines is analogous

o that between

wo dissimilar

media.

CHAPTERg

WAVEREFLECTIONAND

lationships

etweenhe

electricandmagnetic

ields

oftb

incident, eflectcd,

ndtransmitted

avesn Fig.9-2(b)6

similar

o thosewc develo@

in Chapar

8 for tic volte3ct

andcurrentsof the

corresponding

aveson the ransril

sion ine.

For convenience, edivide our treatment f wave >

flection by and transrnission

hrough

planar

boundarb

into

two

Darts:

n this section

we confineour

to

the normal-incidence

asedepicted n Fig. 9-3(a),

in Sections -2

to 9-4

we will examine he more

eraloblique-incidence

ituation epictedn

Fig.

9-

We will show

the basis for the

analogy between

transmissionline

nd

olane-wave

onfi

urations

o

wemayuse ransmission-line

quivalent odels

or

ing

plane-wave

roblems.

Before

we

proceed with our treatment,

we

should

explain he

relationship etween ays

wavefronts, sboth

will be used o represent

he

gation

f electromagnetic

aves. ray s a ine drawn

represent he

direction of flow of electromagnetic

carriedby the

wave,and herefore t is

parallel

o the

pp

agation

unit vectork andorthogonal

othewavefront.Tb.

I

I

m

lx

h

t

d

f

p

ray epresentationf

wavencidence,

eflection,nd

mission hown

nFig.9-3(b)

s

equivalent

o he

representation

epictedn

Fig.

9-3(c).

The wo represo

tations

arecomplimentary;

he ray representations

to use

n

graphical

llustrations,

hereashe

representation

rovidesgreaterphysical

nsight

when

aminingwhat happenso awavewhen tencounters

continuousoundary.

oth epresentations

ill be

our

orthcomin discussions.

9-1.1 Boundaryetween

osslessedia

The

planar

boundary ocated

at

z

=

0 in Fig.

separates

wo lossless,homogeneous,

ielectric

medil

Medium 1,

defined or

z

I

0, s characterized

y

(er,

,rrl

andmedium2, defined

or

z

>

0, is characterized

(s2,

p2).

In medium ,

an ncident r

polarized

lanewa*

with ields

Ei

Hi) is traveling

n

direction

t; : i



9-I

V/AVEREFI-ECTIONAND

RANSMISSION

TNORMAL

INCIDENCE

3 1 3

(D,

Kay representatron

of (c)

Wavefront

representation

of

oblique

incidence

oblique

iniidence

Figure 9-3: Ray representation f wavereflectionandhansmissionat (a) normal ncidenceand (b) oblique ncidence,

and

(c)

wavefront

representation

f oblique

incidence.

medium

. Reflection

nd ransmission

t the

discontin_

uous

boundary

esult

n a reflected

wave

Er,

H,) with

k,

:-

-h

in

medium

1

anda

transmitted

ave

Et,

Ht)

with

81

i in

medium

.

On he

basis

f the ormulations

developed

n

Sections

-3 and

7-4

for characterizins

he

6elds

fa TEM

wave.

he

hreewaves

anbe

describid

n

phasor

orm

by

Incitlent Wave

E'(z)

Et17y

i96r-ir'..,,

fr,(,)

i

"

E'({)

:

9

4

r-ir,,

Thequantities [, E[, andEl are, espectively,heam_

plitudes

f

the ncident,

eflected,

nd ransmitted

lectric

fields,

ll

specified

tr

:

0 and

z

=

0

(the

boundarv

e-

tween

he

wo media).

hewavenumber

nd nrrinsic

m-

pedanceofmedium

arekl

:

aafiOrandnt:

Jn;Ei

and,

similarly,

k2

=

oJEd

and

42

:

J/1fi

for

me-

dium

2.

Theamplitude

6

s elated

o

hesourceresponsible

or

generating

he ncident

wave,

nd

herefore

t is assumed

to

be

a known quantity.

Our

goal

is

to relate

E[

and Ei

eachoT,f,.

We

doso

by applying

oundary

onditions

oi

E andH atz : 0.Accordingo TableG2, the angential

component

f E

is always

ontinuous

cross

boundary

Transmi

ed Wove

(9.3a)

(e.3b)

- J k t z

F r aT r - i

H ' ( z t : 2

x : - l : :

l t 3 "

t k r z

t l t

4t

Reflected

Wave

EtQl:iB5"i*,,,

- i , , " t F l

H'(z) = (-i) x :-:: l = -i -: i ?t{r'

q t

-

4 l

:

iEoe

(9.1a)

(e. l

b)

(9.2a)

(e.2b)

(a)

Normal

incidence



314

CHAPTER9

WAVE

REFLECTION

AND TRANSMISSION

9- l

c/

- - n

(a)

Boundary

betwecndielectric

mcdia

. - n

(b)

TransmissionJineanalogue

Figure

9-4:

The two dielectric

media separated y the

r-y

plane

n

(a)

can

be represented

y the transmission-

line analogue n

(b).

betwe€nwocontiguous edia, nd n theabsencefcur-

rentsourcestthe oundary,

hetangentialomponentof

alsoscontinuouscrossheboundary.Inhe

presentcase,

both

E andH

ofthe

normally

ncident

wave

re angential

to the boundary. onsequently,inceno freecharges r

currents xistatheboundary,he ields fthereflectedand

transmitted

aveswill

have angentialomponentsnly.

In

Fig.

9-4(a) ndcorrespondinglyn

lgs.

(9.2J)

nd

9.3a),

wearbitrarily hosehedirections fE andEt

o coincide

with hedirectionof '

alongthe

ositiver-direction.

heir

truedirections,

elative

o theassumedirections, ill be

determinedy hepolarities ftheamplitudes [ andE[.

As wewill see hortly, oth hemagnitudesnd

polarities

of these

wo amplitudes re

governed

y the values

of

tb.

intrinsic mpedances f the wo media

ry

Nd

nz.

Thc

total clectric icld Er

(z)

in mcdium I is the

sum

d

thc electric iclds

of the nci&nt and eicctcd waves,

md

a

similar statement pplies o thc magnetic

ield

fr11s;.

Hcnce,

Medium

E r ( z ) : E ( z ) * E ( z )

=

i(Etoe- ikn E[eik tz

,

Hr(0)

:

H2(0) or

Simultaneousolutionsor EdandE[ in

terms f

El

grrt

t'i)

,D

I

Hr(z) : I I (z )+ I I (z )

:

g

L

6'o"-itr,

-

Eroeikrzl.

E'o

-

4 l

" 0 _

tl l

(e.40

(9.4b)

(9.5r)

(e.5b)

(9.6a)

(9.6b)

(9.7r)

(9.7b)

Th

llx

cl

bo

dr

rg

?r

@l

a:

fo

Withonly the ransmitted avepresentn medium , hc

fieldsare

Mediutn

E

k)

:i,'(z) :

i6[2-itczz,

,.fr2(z)

ir(z)

i

4,r-(ir,'

At theboundary

z

:

0),

the

angential omponents ftb

electricand magnetic ields arecontinuous.

Hence,

Er(0) E2(0)

or

E;

+

ti

:

4,

" : ( f f i )E;=fE;,

"s= ( ;h )E i : rEL ,



9-I

V/AVEREFLECTIONAND

TRANSMISSIONAT NORMAL INCIDENCE

315

|

":5

-q2-u

(normal

ncidence),

9.8a)

]

" d

q 2 + n l

]

,=4

=

Lry-

(normat

ncidence).

(9.Eb)

EL

4z lu

where

Thequantities

and

z are alled he ef ection oefficient

andtra

smissia coefr c ent, rcsFtively.

For ossless i-

clectricmedia,

?r

and

42

are eal

quantities;

onsequently,

both

andz are ealalso.As

wewill seen Section - .4,

the xpressions

iven

y Eqs.

9.8a)

nd

9.8b)

reequally

applicable

hen

he

media

re

onductive,ut n hatcase

\

and

q2

may be complex, and

hence and z may be

complex swell. FromEqs. 9.8a)and 9.8b),t canbe

easilyshown

hat I and r are ntenelatedby the simple

formula

r

:

1* f

(normal

ncidence).

(9.9)

Fornonmagneticmedia,

no

n r : -

JE,,

'tlo

t/

€"

*

here

46

s the ntrinsic mpedance f

free space,n which

:aseEq.

(9.8a)

may be

rewrittenas

9-1.2

Transmissi0n-LineAnalogue

The ransmission-line onfigurationshown

n

Fig. 9-4(b)

consissof a osslessransmission

ine with

characteristic

impedanceZsl,connectedatzotoaninfi

itelylongloss-

less

ransmission

ine with characteristicmpedance 62

The

nput

impedance f an nfinitely long ine is equal o

its

characteristic

mpedance. ence, t

z

:

0, the voltage

reflectioncoefficient

looking

toward he boundary

rom

the

vantage

oint

ofthe

irst ine) s

_

Z n - Z u r

Zoz Zor

which s denticaln form o Eq.

9.8a).

o show he

basis

for

theanalogy

betweenhe

plane-wave

nd ransmission-

linesituations,heexpressionsforthetwocasesre

iven

n

Table9-. Comparisonofthetwocolumnshowshat here

is

aone-to-one

orrespondenceetween

he ransmission-

line

par_ameters

V

,

I

,

P,

Zd

andthe

lane-wavearam-

eters

E,

H,

k,

4).

This correspondencellowsus o use

the echniques

e

developed

n

Chapter ,

ncluding he

Smith-chart

method or calculating mpedanceransfor-

mations,o solve

lare-wave ropagationroblems.

Simultaneous

resence

fincidentand eflected

aves

in amedium,

uchasmedium n Fig.94(a),

givesrise

oa

standing-wave

attem.

y analogy ith

he

ransmission-

line case, hestanding-wave

atio in medium is

given

by

^ l E r l l n a j .

l + l f l

J

:

;=-

:

l - ---- i ; .

(y. l ) )

lE l lm in

I

-

l r I

|

=4-

E

(nonmasnetic

edia).

9.10)

JE;

+

J%

If the

wo media aveequal mpedances

4; =

42),

hen

|

:

0 andS

:

l, and

f

medium is a

perfect

onductor

with

42

0

(which

sequivalenttoshorl-circuitedrans-

missionine), hen

:

-landS:

oo.Thedistances

from the boundary

o where he magnitude f the

elec-

tric field ntensityn medium is at a maximum, enoted



9-1

WAVEREFLECTIONAND TRANSMISSION

AT NORMAL INCIDENCE

317

which

s analogouso Eq.

(8.84)

or the osslessrans-

missionline

ase. he irst erm

n

Eq.

(9.18)

eprcsents

thc

avcrage

power

dcnsity of the incident

wave,

and

he

sccond

erm

(proportional

o

lf l2)

epresentshe average

power

densityof the eflected

wave.Thus,

S-,=Sl ' *S1" ,

(9.19a)

r p i 1 2

t ^ ' : 2 # '

s!":

n1r1' !%I

- l f

l2sl".

=

n.

fr'rP-i

k'

x

gf

#

","'f

S",,(z):

lmelErlzl

frjtz)l

Even

hough is

purely

eal

whenbothmediare ossless

dielectrics,

e chose o treat

t as

complex,

hereby

ro-

viding n Eq. 9.19c) nexpressionhat s also alidwhen

medium

s conducting.

The average

ower

density f the ransmitted

ave n

medium

s

This

result s as expected rom considerations f

power

conservation.

EffiE

RtdarRadomrllrsisn

A IGGHz aircraft radarusesa narrow-beam canning

antennamounted n agimbalbehinda dielectric adome,

asshown n Fig. 9-5 Even hough he adome hape

s far

from

planar,

t is approximately

planar

over

the narrow

extent of the radar beam.

If

the

radomematerial is a

lossless ielectricwith

p.

:

1 ande.

:

9, choose

ts

thickness

such hat he radome ppearsransparent

o

the

radar beam. Mechanical ntegrity requiresd to be

greater

han2.3cm.

Solution:

The

propagationproblem

is shown in

Fig. 9-6(a) at an eKpanded cale.

The

incident

wave

is approximated s a

plane

wave

propagating

n me-

dium

I

(air)

with intrinsic mpedance

o,

the radome

(medium

) s

of

thickness and ntrinsic mpedance

.,

and

medium 3 is semi-infinite

with

intrinsic

impe-

dance

46.

Figure9-6(b) s

qp

eguivalentransmission-

line model with z

:

0 selectedo coincide

with the

outside urface f the radome, nd he oad

mpedance

Zr

:

4o

representshe

input

mpedance

f the semi-

infinitemedium.

Antennabeam

Dielectric

radome

Antenna

Figure

9-5: Antenna beam

"looking"

through

an aircraft

radome

of thicknessd

(Example

f-i7.

I D r t z

^ ,

. ,

l L O l

=

zlt

l-

-:-

znz

(9.19b)

(9.19c)

(9.20)

Thisexpressionsapplicable henboth edia re ossless,

as

well

as

whenmedium isconducting

nd nlymedium

is

ossless.

Throughthe seofEqs.

9.8a)

nd

9.8b),

tcanbeeasily

shown

hat or ossless edia

for

which and are eal)

t 2 1 - f 2

(losslessmedia),

(9.21)

tl2

q l

which eads o

S"u,

:

Suu,



318

CHAPTER

9 WAVE

REFLECTION

AND

TRANSMISSION

v

Ircidcnt

wave

................

Medium

I

(air)

,lo

Transmittcd

wavc

Medium

3

(air)

tlo

Zt=

4o

z = - d

z = 0

(b)

Figure 9-6:

(a)

Planarsertion

of the radome

ofFig.

9-5 at

anexpanded

cale

and

b)

its transmissionJine

equivalent

model

(Example

9-l

).

t l

t l

z = 4 z = O

(a)

v

t l

, Ltnez ,

isfy

both

the no-reflection

and the

mechanical

nt€grity

requrrcmcnE.

I

Ylllor

Ll0hl

ncldcnt

upon Gl$3sorhcc

A

beamof

yellow

light with

wavelength

f

0.6

pm

ir

normally

ncident

n air

upona

glass

urface.

fthe

surface

is

situated

n the

plane =

Qxnd

hg relativepermittivity

ofglass

s 2.25,

etermine

(a)

the

ocations

f he

electric

ield

maxima

n

medium

(air),

(b)

the standing-wave

atio,

and

(c)

the

raction

ofthe

ncident

power

ransmitted

nto

he

glass

medium.

o

Solution:

a)

We

begin

y determining

hevalue

f

ry1,

,

and :

tt"t

V

t r

-

120r

(9),

ltt, | 1,,

I

= / - . - -

'

\ e , V r o

J + -

&

sl

lr

tr

b

lo

t f

l2Otr

.,,o.%

802

(O),

p

gr

I

tir

fo

m

!

at

-

T

T

,

Requiring he adome o "appear"transparento the n-

cident wave

simply means hat the

reflection

coefficient

mustbezeroat

z

:

-d,

thereby

achieving

otal

ransmis-

sionofthe incident

power

nto

medium

3.

Since21

=

4e

in Fig.

9-6(b), no reflection will

take

place

at

z

=

-d

if

Z1n

46,

which

can be realized

by choosing

:

.z/2

Isee

Section -7.4],wherel2

is hewavelength

n medium

2

and n is a

positive

nteger.

At 10

GHz, the

wavelength

n

airis,l,6

=

clf

=

3 cm, andin

the radome

material

.

L6

3cm

l "z :

Je ,

=

_J_ :1 ._ .

Hence, f we

choose

--

fl"2/2

:

2.5 cm,

we will

sat-

f

=!]:

rJ.:

9:]2=0"

:

-0,.

4z*

41

80it

+

lz0n

Hence,

fl

:

0.2

and0r

:

n. From

Eq.

(9.16),

he

electric-field

agnitude

s a maximum

t

,

0,)'t ),t

.tnax

7;

+

nT

= - - : + n :

4 2

h

=

0 , 1 . 2 ,

. . )

with

),1

Q.f

g.11y.

(b)

^

l + l f l

l + 0 . 2

\ - - - - - - - - - - : - - = _ - r <

r

-

l r l

|

_o .2

(c)The fractionof the ncidentpower ransmittednto the

glass

medium

s

equal

o the atio

of thetransmitted ower

Po

€0

&



9.I

WAVEREFLECTION

AND TRANSMISSION

ATNORMAL

INCIDENCE

319

z = 0

(b)

Transmissionline

nalogue

Figure

9-7:

Normal

ncidence t

a

planar

oundary €-

tween

wo ossymedia.

where

l

:

ar

t

jft,

lz:

az

*

jfz,and

f

_

,rc2 'rcl

tlc2

*

q",

(9.24a)

(9.24b)

n, ,

r : l + l

4c2

-f

tlst

Because

",

and

4",

are,n

general,

omplex,

and may

becomplex

s

well.

Emmple

-3 Normalncidence

na tletal

urlace

A l-GHz

r-polarizedTEM

wave raveling

n the

+z-

direction

s ncident

n airuoona metal

urface oincident

density,

iven

by

Eq.

(9.20),

o the ncident

power

density,

9^"

=

lE[12

2th:

S"",

, I

E'"12

- - - -

= t - :

S;" 2rtz

In

view fEq.

9.21),

lwl:,,t'

L

znt

J 42

S""z

(i

:

I

-

lFl2

|

-

(0.D2

0.96,

or96vo. t

.-*r@

91.4

Boundary

elweenossy edia

In Section -

.l we considered

plane

wave n

a ossless

:nedium

ncident ormally n

aplanar oundary

f another

-ossless

edium.Wewill nowgeneralize

urexpressions

:o ossymedia.n a mediumwithconstitutivearameiers

s,

p,

o), the

propagation arameters

f interest

re he

:ropagation

onstanty a

+

jP

^ndthe

omplexntrin-

'ic

mpedance

".

The

general

xpressionsor a,

p,

and

4"

rre

given

by Eqs.

7

121a),

7

121b),

nd

7.125),

espec-

:r'ely,andapproximatexpressions

re

given

n Table

-2

ror he

special ases

flow-lossmedia nd

good

onduct-

urg

media. f medium1 is characterized

y

(e1,

plr,

o1)

andmedium2 by

(ez,

p2,

o2),asshown

n Fig. 9-7, he

expressionsor theelectricandmagnetic

ields n media

and2can eobtainedromEqs.

9.11a)

hrough

9.14a)

f

Table - byreplacing

&

with

7

and

4

with

4"everywhere.

Thus,

lletlitun

l

Etk\

:

*Eik-v'z

+

lev'z),

r i

f r , i31

=

!191s-ztz

leYt '

,

4" t

EzQ):fu

Eie-nz,

fr'e) y, i "-n,

(9.22a)

(9.22b)

(9.23a)

(9.23b)

\c z



with he.r-y

plane

t

z

:

0.If theamplitude f theelec-

tric

field of the ncidentwavc s

12

mV/m)

and

he metal

surfacc s madeof

coppcr with

p,.

-

1, er

:

1, a16

o

:

5.8

x

l0?

(S/m),

obtainexpressions

or the nstan-

taneous

lectric and magnetic ields in the air

medium.

Assumehemetalsurface o be scveral kin depths eep.

Solution:

n medium

(an),

a

:

0,

CHATTfER9 WAVEREFLECTIONAND

Widr

Eb

:

12

(mV/m),

the instantaneousields

sponding

o these

hasors

re

ELk,i:n

IEtk)ei''tl

:

i

2Ei

sin&rz

ina.rr

:i24sin(20n2/3) sin(22 x l0er) (mV/m\

IJt(z, t)

:

nealJt(z)eta'7

E l

=

j'219

cos 12cos r;r

4 1

=

j

64cos(2ln

z/3)

cos(22 x 10er)

(ptA/nt

Plotsof the magnitudeof E1

2,

I

)

and H

I

(2,

t) are shosr

in

Fig.

9-8 as

a function of negative

x

at various

valtn

of a)t. The standing-wave

pattems

exhibit a repetitic

period

of

^/2.

Md

E and H are

in

phase quadmqtrc

(90' phasetshift) n both spaceand time. This behavic

is identical with that of the standing-wave

pattems

c

voltage

andcurrenton a shorted ransmissionine.

I

REVIEWUESTIONS

Q9.1

What

boundary onditions

ere

usedn theded-

vations

f theexpressionsor f and ?

Q9.2

In the adar adome esign f Example -

,

all hc

incident nergyn mediurn ends pgetting ransmittcd

into medium

3,

andvice

versa.

Does his mply thatoo

reflections

ake

place

withinmedium ?

Explain.

Q9.3

Explainon hebasis f boundary onditions

hy

it is necessaryhat

:

-1

at the

boundary

etween

dielectricanda oerfectconductor.

EXEBCISE.1

Toeliminatewave eflections,dielectric

slabofthickness and elative

ermittivity

.,

s o

be

n-

serted etweenwo semi-infinitemediawith relative

er-

mittivities€n

:

I andsr.

=

16.Use he

quarter-wavc

^ r ^ O

(D

Zlf

X LV'

I J : k t

=

c 3 x lOE

, t L - t t u - J " \ r . ) ,

20n

3

(ra(vm),

2n

-,

=

U.J

m.

K1

€ o

e' a€€o

At

/

:

1GHz,coppers anexcellent

onductor ecause

5.8

x

107

=l

x

l0e>>.

2nx l }ex ( l 0 -e /36n )

Use fEq.

7.132c)

ives

r t c z = ( l +

)

:8 .25 ( l

+

i )

Since

4.,

is sosmallcompared o

46

:

377

(Q)

for air, he

coppersurface

cts, n effect,like a shortcircuit. Hence,

^ 0 c " - 4 o

:

- ' - t - r

rlc|

*

49

Upon

setting

:

-

1 n Eqs.

9.

I a)and

9.

2a)

of Table

9-1,

wehave

Ere)

--iEik-jkrz

-

"ikvy

:

-?j2Eisinktz,

F i

f r r(z)

i

'o

1e-ikrzyslkft1

ry l

^^EI

: y Z - COS tZ .

4 l

: ( l + j ) [

ft

t /2

1l

0-

I

x 1

0e

x4n

53

r' 10

(ma).

f

h

;-

II

nl

-ll

i =

(9.25a)

(9.2sb)

gI

Lc

u

--D

.{r

itflL

o



322

CHAPTER

9 WAVE

REFLECTION

ANDTRANSMISSION

Figurr 9-9: Wave reflection and refraction at a Dlanar

boundary between

different

media.

A,O'

and AlO', as shown

n Fig.

9-9. The

incident

and

reflected

waves

propagale

n medium

I with

the

same

phase

velocity

upr

:

|

/

J

1,t

1,

nd

the

ransmitted

wave

n me-

dium

2

propagates

ith

a

velocity

zo2

:

l/Jlt2\.The

time it

takes

he

ncident wave

o

travel

from

Ai

to O, is

the

same

as he time it takes

he reflected

wave

to ravel

from

O to A. and also the

time it takes

he transmitled wave to

travel

from O to Ar . Since

ime is

equal

to distance

divided

by velocity,

t follows

that

AiO'

_

OT,

_oT,

uPr

uPr

up ,

From he

geometries

fthe

three ight

triangles

n

Fig-

9-9,

we

deduce hat

Use

of these

xpressions

n

Eq.

9.26)

eads

o

,i:4 (Snell's

aw

ofreflection), (9.2ga)

sin

Q

up,

ttt,tt

t

sin01 lh ,,1t"r"z

(Snell's

aw

of refraction). (9.2gb)

Snell's

aw

ofreflectbn

states

hat

heangleofreflectionis

equal

o he

angle

fincidence,

nd

Snelt\

lnw

of

refrao

doz

provides

a relation

between

in

g,

and

sin

e

in

rcnnr

of the

atio

of

the

phase

elocities.

The ndex

of refraction

of

a medium,

n

is defined

as

the

atio

ofthephase

elocity

n free

space

i.e.,

he

speed

of light

c.1

cthe

phase

elociry

n

rhe

medium.

hus.

C

u n

(e.2e)

In

view

ofEq.

(9.29),

q.

9.28b)

may

be ewritten

s

i

.

d

I

I

sin

4

sind;

n t

I u . . .

=

r : . / - .

( 9 . 3 0 )

nz

I

F,2e1,

For

nonmagnetic

aterials,.r =

&r:

:

l, in

which

ase

A i O ' : O O ' s i n 0 i ,

d4:dasino,,

O

lr:

6

g'

"in

r.

(9.26)

(9.27a)

(e.27b)

(9.27c)

sin4 nt

lE: n)

* ' o ' = " r = / ; : ;

( f o r 4 1

t t z \ '

1 B ' 3 t )

where

4

=

"/pf!

is

the

ntrinsic

impedance

f a

dielec-

tric

medium.

Usuatly,

materials

ith

higher

ensities

ave

higherpermittivities.

Air, with p.

:

a.

=

l, has

an ndex

of refraction

no

:

l.

Since

for

nonmasnetic

materials

n

-

./+,

a

material

is

often refcrred

ri

as ,ore dense

tllan

a

secotld

material

if the

indes

of refraction

of

the

fr,st

moterial

is greater

than

that

of the

second.

At normal ncidence Ai : 0), Eq. (9.31)gives91= 0,

as

expected,

and

at oblique

incidence

4

< di

when



324 CHAPTER9 WAVEREFLECTIONAND

TRANSMISSION

Substituting

q.

9.33)

nto

Eq.

9.34)

ives

:

sin0r.

Hence, r

:

gr.

The slabdisplaceshebeam's

osition,

but hebeam'sdirection emains nchaneed. I

EXERCISE.4

In

the

visible

part

of theelectromagnetic

spectrum,he ndexof refraction f water s 1.33.What s

the critical angle or light

waves

generated

y an upward-

looking underwater

ight source?

Ans.

9.

:

48.8'.

(See

O)

EXERCISE.5

Ifthe ightsource fExercise .4 ssituated

at a depthof

I

m below

hewatersurface nd

f its beam s

isotropic(radiatesinalldirections),

owlargeacirclewould

it illuminate hen bservedromabove?

Ans.

Circle's iameter:2.28m.

(See

S)

9-3 Fiber ptics

By successiveotal ntemal eflections, s ndicatedn

Fig.9-12(a), ight canbe

guided

hrough hin

dielectric

rodsmade fglass r ransparentplastic,nown

asopdcal

./fiDers.ecausetheight sconfinedotraveling ithin he

rod, heonly

oss

npowerisdueo eflections tthe ending

and eceiving nds f the iberandabsorption

y he iber

material

becauseitis

ot

a

perfect

ielectric). iberoptics

is useful or the ransmissionf wide-bandwidth

ignals

and

n a wide ange

f

imaging

pplications.

An optical iberusually onsists f a cylindicalfiber

core

with an index of refraction n

1,

surrounded y an-

other cylinderof lower ndex

of

refraction,

", called

a

cladding,

as shown n Fig.

9-12(b).

The

cladding

layerserveso optically solate he fiber rom

adjacent

fiberswhena largenumber f fibersarepackedn close

proximity,

hereby voidinghe eakage flight from

one

,'n

=

fr)*).'*

fiber

to another.To satisfy he condition of

total nternal

reflection, he ncidentangle93 n

the fiber core must

bc

equal

o

orgeatcrthanthecritical

angled" ora wave

n

ttc

fibermedium(with

zs) ncidentuponhe

claddingmediug

(with

n"). From

Eq.

9.32a),

ehave

sin

0"

(9.35)

To meet

he total-reflection equirement hat

0:

>

0c,

t

is

thennecessaryhatsin03

2

nrfns.The

angle 2

s

thc

complement

f angle 3,andcos92

:

sin0:. Hence,

he

necessary

onditionmay bewritten as

t

(e.36)

Moreover,

d2 s related to the incidence

angle on the facc

ofthe fiber,Pi,by

Snell's aw:

sin 92

:

sindi ,

(9.37)

where

n6 s he ndexofrefractionofthemedium

surround-

ing the iber

(n6

=

I for air andno

:

L33 if the iber s

n

water).or

"orB,

:

|-t

-

L

UsingEq.

(9.38)

n

the eft-handsideofEq.

(9.36)andthen

solving or sin 9i

gives

nc

nf

cos02

a

fl"

n f

nO

nf

( 'o) ' r ,n,

ol ' "

.

(9.3s)

\ n f

, /

I

The

acceptance

dngle

du s defined

as he maximum

valuc

of

Q

for which the

condition of total intemal

reflecdon

remainssatisfied:

s i n d ; :

@ ! - n f 1 t t 2

sin4 = l1r; - nf,)t/2. (9.40)

(e.39)

T

r

d

b

m

cl

!-O

r

d

fq

tr

aa

cb



326

CHAPTER

9

WAVEREFLECTIONAND

TRANSMISSION

9

side of

the fibcr, t is necessaryhat r be shorter han I.

As a safety

margin, t is common

practice

o require hat

T

>

2t .

The spread-outwidth r is equal o the time delay Ar

between he anival of the slowest ay and he astest ay.

The slowest ay is the onetraveling he ongestdistance

and

correspondso the ray ncidentupon he nput aceof

the iber

at he

acceptance ngle

6a.From he

geometry

of

Fig.9- 2(b)andEq.

9.36),thisraycorrespondstocos

2

nc/nf.

Foran

optical iberoflength

, the ength fthe

path

traveled

bv such

a

rav s

I{lg&*l,.

TransmisslonalaR8l6

onoplical

Flisrs

A l

-km-long

optical iber

(in

air) s madeofa fiber

c6p

with

an ndcx of refractionof I .52 anda claddingwith

an

indexof rcfraction

of L49.

Determine

(a) theacceptancengle0",and

(b)

themaximumusable ata ate hatcanbe

ransmitted

through he iber.

Solution:

a)

From

Eq.

9.40),

I

sin0"

-6u' ! -

nz;t /2

111.52)211.49;21r12

9.3,

n O '

which

correspondso 0a

=

17.50.

(b)

From

E{

(9.45),

l n e

I

-

_

-

I ___

r m ( -

COSU2

nc

e

b

h

oI

a[

O

til

p,

ti

pl

l

is

r

tn

c

p

p

ui

t

p

al

tc

dl

e

c

t

and

ts travel time in the fiber at the

velocity

uo

--

c nlis

(e.42)

Theminimumtime f travels ealized v heaxial avand

is

given

y

2 x 103 1.52(1.52

1.49\

EXERCISE

.6 If the ndexof refraction

f the cladding

material

n Example9-5 s increasedo 1.50,what would

be henewmaximumusabledata ate?

Ans. 7.4

Mb/s). (See

G;

9-4 Wave eflectionnd ransmissionl

0bl iquencidence

For normal incidence, he reflection

coefficient

f and

transmission oefncient r of a

boundarybtween

two

differentmedia s independent

f the

polarization

f the

incidentwave,

becausehe electricandmagnetic

ields

of a normally ncident

plane

wave

are both always

an-

gential

o the boundary egardless f the wavepolariza-

tion.This s

not

hecaseor obliquencidence t an

anglc

lmax

_

lni

up

cn c

I

T

(fr

')

(s)

e44)

(9.41)

(bits/s). (9.45)

2lnr(nr n")

3 x 1 0 8 x 1 . 4 9

:

4.e

MbA).

r

(9.43)

The total time delay

is therefore

t

:

Lt

:

l6a1-lnt;n

:

As

we

statedbefore,

o retrieve he desired nformation

from the transmitted ignals,

t

is advisablehat f, the

interpulse

eriod

of

the nput rain of

pulses,

eno shorter

than2r. This, n tum,meanshat hedata ate

in

bits

per

second),r equivalentlyhe

numberofpulses

er

second,

thatcanbe ransmittedhroush he iber s imited o

I

-

nf .

c

- l

I o - - i

I

2r

Zlns(n1 n")

h

d

cl

d

d

lt



94 WAVEREFLECTIONANDJRANSMISSIONATOBLIQUEINCIDENCE

J Z I

O,

+

0. A

wave

with

any

specifiedpolarization

mav

bedescribed

s he

superposition

f

two

orthogonally

po_

larized

warrs,

onewith

ts clectric

icld paralel

totheilane

of incidencc-and

it

is

calld

paraltct

polarization-__$nd

another

with

is

electric

field perpendicular

o the plane

of incidence-and it is calledparpendicular polariza-

tion.

The

plane

of incidence

s defned

as

the

plane

con_

taining

he

normnl

o

the

boundary

nd

the

irection

of

propagation

of

the

incident

wave-

T-ltese

wo

polariza_

tion configurations

re

shown

n Fig.

9-14.

n which

tre

plane

f incidence

s coincident

ith

the

t_z plane.

o_

larization

with

E

perpendicular

o

the

ptane

oflncidence

is also

called

transverse

electric (IE)

polarization

be_

cause

E is perpendicular

o the plane

of

incidence,

nd

that with

E

parallel

o

the

plane

of

incidence

s

called

transverse

magnetic

(IM)

polarization

because

n

this

caset is themagneticieldthat s perpendicularo the

plane

fincidence.

Instead

of

solving

the

reflection

and

transmission

problems

or

the

general

ase

of

a wave

with

an

arbi_

trary polarization,

t is

more

convenient

n

practice

o

first

decompose

he

incident

wave

(Ei,

Hi) into

a

per_

ryngicu]arly

olarized

omponenrEi,

Hl)

ana

ar_

allel polarized

omponent

8i,,

Hi,),

and

ien

after-de_

rermining

he

reflected

wavei

1E_,',

l)

and

(Ei,

Hi

)

due

o the

wo

ncident

omponents,

hJreflected

aves

can

be added

ogether

o give

the

total

reflected

wave

conesponding

o

the

original

ncidentwave.A similarprocess

pplies

o the

ransmitted

wave.

9-4.1

Perpendicularpolarization

In

Fig.

9-15,

we

show

a

perpendicularly

olarized

nci_

dent

lane

wavepropagating

long

he

;-direction

n

di_

electric

medium

.

The

electric

ietd phasor

E1 point,

along

he.y-direction,

nd he

associated

agneti"

fi"la

phasgr

Hi. is along he y1-axis. he directions f El

and

H!

satisfy

he

condition

hat

E! x

fri points

alon!

(a)

Perpendicular

olarization

z = 0

(b)

parallelpolarization

l'igur€

9-14:

The plane

ofincidence

is the plane

contain-

ing

the

direction

of wave

travel,

R, and

the

surface

normal

to

the

boundary

which

in

the present

case

s the

plane

of

thepaper.

A wave

s

(at

perpendicularty

polarizid

when

its

E

is

perpendicular

o the plane

of incidence

and

ft)

parallel

polarized

when

ts

E

lies

n theplane

ofincidence.



94

328

CHAPTER

9

WAVE

REFLECTION

AND TRANSMISSION

R

the

propagation

irection,1. Theexpressionsor

sucha

plane

wave re

given

by

(9.46a)

(9.46b)

Substitutingqs.

9.47

) and(9.47b\into

qs.

9.46a)ad

(9.46b)

ives:

Incident

Wave

tr: g nior-io"' ,

q i ,

- i

E lo " - ; r , ' ' .

- " n t

E',

--

9

Elor-

ior,'

sinei+zcosdi),

fr i

=

1-icosa;

2sind1)

- i

x

:]q

.,-ilr

(r

sin

4+z

cos9,)

nr

c/l

st

rn

tcl

P

d

si

tn

c

br

q

b

6

n

di

|h

!0

F

where

Eio is the amplitude

f the electric ield

phasor

at .ri

:

0,

h

=

-Jpld is

the wavenumber,

nd

qt

:

Jp;/e

I

is the intrinsic mpedance,

oth or me-

dium .FromFig.9-l5,thedistance.riandtheunitvectori,

maybeexpressedintermsofthe

.x,

y,

z)

globalcoordinate

system

s ollows:

ti

:

-r sin

4

+ z

cos

9i,

(9.47

)

i i

:

- icos9i

+

isin0i .

(9.47b)

With

the aid of the

directional elationships

ivenh

Fig.9- 5 for he eflectednd ransmitted aves,he ielL

are

sivenbv:

(9.4&)

(9.48b|



330

CHAPTER9

WAVEREFLECTION

AND TRANSMISSION

9'

through

consideration

f the

ray

path

ravers€d

y the n-

cident.

cflected,

nd ransmitted

wavefronts.

In

view ofEq.

(9.54),

heboundary

onditions

given

by

Eqs.

9.51)

nd

9.53)

educeo

Ero+ E\o:

E

Lo,

(9.s7a)

to '4(-r , ,"

+

EI^)

=

-totQ

E,,,n.

9.57b)

nr

-"

42

Thesewo

equationsan esolved

imultaneously

o

yield

the

ollowingexpressions

or

the ef lection nd ransmis-

sion

coefficientsn the

perpendicular

olarization

ase:

Ftf

zio

r t

- IU

f2

cos

0;

-

[1

cos01

q2cosqi

4r

cos4

'

2112

os0i

4,

cosdt

+

4l

cos r

(9.58a)

(9.58b)

These

wo coefficients,

hich ormally

areknownas he

Fresnel

eflection nd ransmissiott

officientsfor

per-

p

endicular

p

o

arization,

are elate.dby

z r : 1 * f r .

( 9 . 5 9 )

If medium s

a

perfect

onductor

ar

:

0), Eqs.

9.58a)

and

9.58b)

educe

o fa

:

-

I and

z1

=

0,

respectively,

which meanshat he ncidentwave s totally reffected y

theconducting

edium.

For nonmagnetic

ielectrics

ith

pr

:

p2

=

po

and

with thehelp

ofEq.

(9.56),

heexpression

or fr canbe

written as

Since

ez/er)

-

@z/n)",

this expressionan

also

bc

written n terms

of the ndicesof refraction

z1

and

n2.

:ru

w.Yrlncldoilobllqu.ty

onNSollSur{rGe

Using hecoordinarc yscm ofFig. 9- 15,aplancwaw

radiated y

a distantantennas

ncident n air upon

a

planc

soil surfacc t

z

:

0.

Theelectric ield ofthe ncident

warrc

is

given

y

E

-

il0Ocos(or

nx

-

1.73trz)

(V/m),

(9.61)

and he soil

mediummay be assumedo be

a ossless

ic-

lectric

with a rclative

permittivity

of 4.

(a)

Determine

r, ft2, nd he ncidence ngle i.

(b)

Obtain

xpressions

or the otalelectric ields n

air

and n the

soilmedium.

r

(c)

Determine he average

ower

densitycarriedby

thc

wave raveling n the soil

medium.

Solution:

a)

Webegin y converting

q.

9.61)

nto

pha-

sor

orm,akin o heexpression

iven

y Eq.

9.46a):

fri

-

t l}Oe-iltx-it73.trz

:

i100e-i*t'i

(V/m),

(e.62)

where ; is the axisalong

which he wave s traveling,

and

k l x r : vy *1 .73 r2 .

Using

Eq.

9.47a),

ehave

t1.r;

/<1:rin9;

*

krzcos0i.

Hence,

tr sin

9i z'

h

cos0i : 1 '73n,

which ogether

ive

(9.63)

k1

=

Jrz

|

\1.73nt2

2n

lrad/m).

n

Si

P

m

1

is

tn

dl

U

o i : tan - r (# ) : r '

(e2lE)

-

sin2 i

os8i

-

F .

-

(for pt1

1111.

cos0i

*

(e.60)

(e2le)

-

sin2 i



9.4

WAVE

REFLECTION

AND

TRANSMISSION

AT OBLIQTJE

NCIDENCE

331

The

wavelength

in medium

I

(air)

is

2tr

1 1

- ; : 1 6 ,

f,r

and

he wavelength

n medium

2

(soil)

is

. r r lLz: ---- : = - =0.5m.

Je"

\/4

Theconespondingwave

number n

medium

2 s

- 2 n

kz=

-:4o

(rad/m).

,'2

^ . = ; ,

SinceE'

s

along

,

it is

perpendicularly

olarizedi

is

per-

pendicular

o he

plane

ofincidence

containing

hesurface

normal and he

propagation

irection

1).

(b)

Correspondingo

0i

:

30", he

arsmission

angle

1

isobtained ith hehelpofEq. (9.56):

k r ^ 2 r

sin0r:

-

s in1

=

-

sin

0" 0.25

or

:

t4.5 '

With

e1

:

eoand

2

:

tr2e0

4es, he

eflection

nd

transmission

oefflcients

or

perpendicular

polarization

are

etermined irh

hehelpofEqs.

9.59)

nd

9.60),

cos

01

F r :

= _0.38,

r r : l * I . r : 0 . 6 2 .

UsingEqs.

9.48a)

nd

9.49a)

with

Elo

:

100V/m

and

1i

4,

the otal

electric ield

phasor

n medium

is

El:E,,

+Ei

:

iEloe-ltr(r

sinq+zcosdi)

*

9f

Eloa-

jlr

(t

tin'i-z cos

)

: i lgge-.i

r"

+ t -t3,t

')

?38e- i @x-t.73,t2),

and he

conesponding

nstantaneous

lectric

ield in

me-

dium

is

E!(.r,

,

)

=ne[E\ei^f

=

i[100

cos(arr

ttx

-

l.73ttz)

- 38cos(arr rt ! l.73tt7)l (V/m).

In medium

2,

usingEq.

(9.49c)

with Elo

=

rr Elo

gives

E":

9t

E1o"-

ir'('

sio r+uos

t)

=

t62e-

Qr

+3'8h")

and,

correspondingly,

E!(.r,

,

)

nelfr'r"i'')

- j62cos(at-tx -3.87t2) (V/m).

(c)Inmedium2,

4z no/

Jeh

-

L2On

J4

:

60rr

((.)),

and

he average

ower

density

carriedby

thewave s

I El t2 t<.r \2

S"t"

r".rot

=

;Y+-

=

10.2

an2 z X

OUz

(Wm2).

r

9-4.2 Parallel

olarization

If we interchange

and H of

the

perpendicular

o-

larization

situation,

while keeping

n

mind

the

require-

ment hat

elateshe

directions

f E andH

to the direc-

tion

of

propagation

or each

of the incident,

eflected,

and

transmitted aves,

we

end up with

the

geometry

shown

n Fig.

9-16 for

parallelpolarization.

Now the

electric

ields

ie in the

plane

of incidence,

nd

he

as-

sociated

magneticields

are

perpendicular

o the

plane

of incidence.

With reference

o the

directions ndicated

in Fig.

9-16, he fields

of the incident,

eflected,

nd

transmitted aves regivenby

(e2/e)

-

sin2 1

cosQ

n/(e2/e1)

sin2f l



332

CHAPTER9

WAVEREFLECTIONAND TRANSMISSION

Figurt

9-16: Parallel

polarizedplane

wave ncident at an

angle 0; upon a

planar

boundary.

Incident

Wave

fr,i

9,r[oe-;t'',

:

( icosgl

-

2sin0) Eipe-ikr

(r

sinq+z os i),

(9.65a)

Ell^ E""

ll ',,

=g

-

rtu

"-ir ',,

=

i:]9

?-iIr{rs;q+2"*4,t.

1!.65b)

t '

'

n t

'

n l

Refected

Wave

El,

-

l,tl,o"-to'"

:

(icos4

+

2sin0,)

El,oe-

k't '" i"e'-z os ),

(Q.65c)

: -5,

Eio

,-ro', '

" ine.-!cosd'),

7l l

Transmitted

Wave

E1

1r;o"-;t'"

=

(icosdi

-

isinfl)Ef,oe-ikzG

tin&+zcosq),

(9.6i)

Ft r t

i I, =9 !-lo- -it ', - 9"10"-lrrt'rio4+zco'c). 9.650

t t ' n 2 ' n 2

By matching he tangential

components f fr and

f,

h

the two media at

z

:

O, as we did

previously

n

thc

perpendicular-polarization

ase,

we

againobtain he

rc-

lations efiningSnell's

aws,aswell as he ollowing

ex-

prcssions

or heFre

nel

cflection nd transmission

a

fficients for

parallel

polarizatian:

Eli"

f'

:

--":

Eio

Ft

",t0

Ei,o

(9.66a)

(9.66b)

42

cos

4

-

4l

cos9i

4 z c o s 4 * [ 1

c o s 0 1

2q2 cos0i

4,

cos

Ar

+

4l

cos t

The

preceding

xpressions

anbe shown o

yield

the el}

tron

r n : ( l * f 1 )

cos

(9.67)

cos

We noted earlier in connectionwith the

perpendicular-

polarizadon ase hat,when he secondmedium s aper-

fect conductorwith

rz

=

0, the ncident

wave

gets

otally

reflectedby the boundary.The same s true for

paralld

polarization;

etting

2

:

0 in Eqs.

9.66a)

nd

9.66b)

gives

1

:

-1

and 1

=

0.

Fornonmagnetic aterials, q.

9.66a)

ecomes

-(e2/e)

cosQ

*

(e2le)

-

sin2 ;

(e2 ler )co s4+ (ezle)

-

sin2 i

(for p,1 1tr1.

(e.68)

T

II

rll

r €

of

ts

t$

B,

d

ls

l r

of

H i

= - i

' " €

"

nr

+

T

t !

9.65d)



94

WAVE REFLECTION

ANDTRANSMISSION

AT

OBLIQUE INCIDENCE

333

Per0sndicular

olarizalion

For

perpendicular

olarization,

he

Brewster

angle

o1

canbe

obtained

y setting henumerator

fthe

expression

for f1,

given

by Eq.

(9.58a),

equal o zero

or,

equivalently,

when

?2

cos8i

=

?r

cos

4.

(e.69)

After

(1)

squaring

oth sides

of Eq.

(9.69),

2)

using

Eq.

9.56),

3)

solving

or01, nd

hendenoting

; as

o1,

we have

sin

ds1

:

(e.70)

Beeuse

hedenominator

fEq.

(9.70)

goes

o zerowhen

Itt

:

lr2,

0Br

does ot exist

or

nonmagnetic

atertals.

Parallel

olarizalion

The

value

of 91, enoted

B at which

fl

:

0 can

be

found

ysetting

henumeratorofEq.

9.66a)

qual

ozero.

The result

s

identicalwith

Eq.

(9.70),

but wirh

p

ande

interchanged.

hat

s,

sin

op

-

(9.71)

For

nonmagnetic

aterials,

Io illustratethengularvariations

f he

magnitudesof

1

rnd f1, in Fig.9-17we

show

plots

or waves

ncident

n

rironto hree ifferentypes

fdielectric

urfaces:ry

soil

s.

:

3),wetsoil

e.

:

25),

and

water

e.

:

8

).

Foreach

Dfthe urfaces, ) fr : ft atnormalncidenceft : Q;,

as

expected,

2)

lfrl

:

lfrrl

:

I at

grazing

ncidence

4

:

90"),

and

3)

f

goes

o zeroatan

angleabeled

he

BrcN'ster ngle

n Fig. 9-17.Fornonmagnetic

aterials,

r.he

rewsteranglexists

nly

orparallel olarization,

nd

rts aluedependsontheratio

€2/el

,

aswewill

see hortly.

.lt theBrewster ngle,

he

parallel-polarized

omponent

of he ncidentwave s totall| tran.tmitted

nto

medium .

9-4.3 Brewsterngle

The

Brewster ngle

o

is

defined s he ncidence

ngle

1

atwhich heFresneleflectionoefficient : 0.

rblt

st n

,

_ ]

: t a n '

E

\/

I

+

(€rlr2)

(for

p.1

tt).

(9.72)

e l

The

Brewster

angle s also

called

the

polarizittg

an-

g/e.

This is because,

f a wave

composed

f both

per-

pendicular

nd

parallel

polarization

omponents

s in-

cident

upon a nonmagnetic

urface

at the Brewster

angle o1, the parallelpolarizedcomponents totally

1

-

(pae2/

.2e)

|

-

Q.Lt/tr)z

Wet soil

(e,=25)

Dry soil

( e .=3 )

-

-

- '

l0

20 30 40

50

(0s

dry soil)

(AB

wct

soil)

Incidenceangle d; (Degrees)

Figure

9-17:

Plots or

lfl l

and

fn I

asa unction

fQ for

a

dry soil surface, wet-soil

urface, nd

a water

urface.

Foreachsurface,

fl |

:

0 at theBrewster ngle.

1-

(e1p.2/e2p)

|

-

( t

1 /e)2



Lasers

Lasers

are used

n CD and DVD

players,

ar-code

readers, ye surgeryand

multitudes f

other

sys-

temsand applications.

laser

acronym

or Light

Amplification y Stimulated

mission f Radiation

-

is a source of

monochromatic

single

wave-

length),

oherent

uniform

avefront),arrow-beam

light, n contrast

with

othersources

f light

such

as

the

sun

or a lightbulb)

which

usually ncompass

wavesof manydifferent

wavelengths ith random

phase incoherent). lasersourcegeneratingMi-

crowavesscalleda

maser.

he irstmaserwas

uilt

in 1953by CharlesTownes nd he

first aserwas

constructed

n 1960bv TheodoreMaiman.

TECTINOLOGYBRIEF:

Basic rinciples

Despitetscomplex

uantum-mechanical

an atom can be convenientlymodeledas

a

cleus

containingrotons

nd

neutrons)

by a cloudof electrons. ssociated ith he

or moleculeof any

given

material

s a

set of

quantized discrete)

nergyslales

(

that

the electrons

an occupy.

Supply

of

(in

the form ol

heat,

exposureo intense

ight,

other

means)by

an elternal sourcecan cause

electron o move rom a lowerenergystate o

higher

energy

excited)

tate. Exciting he

is

called

pumping

because

t leads

o

the

population

f electrons n higher states

Spontaneous mission f

a

photon light

o

t

h

b

t

h

l"

t

el

o

h

i,

el



occurs

when

he electronn

he

excited tate

moves

to a

lower

state

B),

and

stimulated

mission

C)

happens

when

an emitted

hoton

entices"

an elec-

tron

in an excited

state of another

atom

to move

to a

lower

state, herebyemitting

a second

pho-

ton of

identical

energy,wavelength,

nd

wave-

lront

(phase).

Principlef0peration

Highlyamplified

timulated

mission

s called

as-

ing.The

asing

medium an

be solid,

iquid,

r

gas.

Laseroperation

s illustratedn

(D)

or

a ruby

crys-

tal

surrounded y a tlash

ube

(similar

o

a cam-

era flash).

A

perfectly

eflecting

mirror

s

placed

on

one end of the crystal

and

a

partially

eflect-

ing

mirroron the

other end.

Light rom

the flash

tube

exciteshe

atoms; ome

undergo

pontaneous

emission,

generating

photons

hat

cause others

to

undergo

stimulated

mission;

photons

moving

along

he

axis of the

crystal

will bounce

back

and

forth

betweell

he mirrors,

ausing

dditional

timu-

lated

emissioni.e.,

mplification),

ith

onlya frac-

tion

of the

photons

exiting

hrough

he

partially

e-

flecting

mirror.

ecause

ll

of he

stimulated

hotons

are identical,

he

ightwave

generated

y the aser

is

of a single

wavelength.

Wavelength

Golor)

fEmitted

ight

The

atomof any

given

material

as unigue

energy

states. he

difierence

n

energy

etween

heexcited

high

energy

tateand he

stable

ower

energy tate

determines

he wavelength

f the

emitted

photons

(EM

wave).Through

proper

choice

of

lasing

ma-

terial,

monochromatic

aves

an

be

generated

ith

wavelengths

n he

ultraviolet,

isible,

nfrared

r

mi-

crowave

ands.



336

transmitted

ltto the

secondmedium,

andonly the

perpen-

dicularly

polarized omponents

reflected y

the surface'

Nahtral

ight, including

sunlight and

ight

gencratcd

y

most

manufactured

ources,

s considercd

npolarized

e-

cause

he

direction

of the electric

ield

of the

ight waves

varies

andomly n

angleover

he

plane

Perpendicular

o

the

direction

of

propagation.

hus,on average

alf of the

intensity

of natural

ight is

perpendicularly

olarized

and

theother

halfis

parallelpolarized.

Whenunpolarized

ight

is incident

upon a surface

at the Brewster

angle, he

re-

flected

wave s strictly

perpendicularly

olarized.

Hence,

the eflection

rocess

cts

sa

polarizer.

REVIEW

UESTIONS

Q9.4 Can total internal eflection akeplace or a wave

incident

n

medium

1

(with

n1)

onto medium

2

(with

n2)

whenn2

> nr?

Q9.5

What

s the difference

between

he boundary

con-

ditions

applied

n Section9-1.1

or

normal ncidence

and

those

applied

n Section

94.1 for

oblique ncidence

with

perpendicular

polarization?

Q9,6

Why

is the Brewster

angle also

called the

polariz-

ing angle?

Q9.7

At the boundary,

he

vector sum of

the tangential

components of the incident and reflected electric fields

has

to be equal

to the

tangential

component of

the trans-

mitted

electric

field. For ".,

:

I and

sr,

:

16, de-

termine

the

Br€wster

angle and

then verify

the valid-

ity of

the

precedingstatement

y sketching

o scale he

tangential

components

of the three

electric

fields at the

Brewsteransle.

CHAPTERg

WAVEREFLECTIONAND

Ans.

f1

-

-0.48,

zr

:

0.52, "[

:

-0.16'

zr

=

0.58.

EXERCISE.8

Detcrminc

the Brewster

angle

for

tlrs

boundary f

Exercise .7.

Ans.

fu

=

63.4'.

(See

O)

EXERCISE.9

Showthattheincident,reflected,andtrary.

mitted

electric

andmagnetic

ields

given

by Eqs.

9

ttuough

9.65f)

all

have he same xponential

hase

tion

along he.r-direction.

Ans.

With thehelp

of Eqs.

9.55)

and

9.56),

all six

are hown

o varyas

€-lrrr

sin4.

(See

(D)

9-5 Refleclivitynd ransmissivity

The eflqrtion

nd ransmission

oefncients

epresent

ratios of the

reflected and

transmitted electric

field

plitudes

to the

amplitude of

the incident

wave. We

examine

power

ratios,

and we start the

process

by

sidering

he

perpendicular

olarization

case.Figure9

shows a circular

beam ofele€tromagnetic

energy

upon

heboundary

etweenwo contiguous,

ossless

dia. The

areaof the spot

illuminated by

the beam

s

and the incident, reflected, and transmitted beams

electric-field

mplitudes

lo, Eio, and

Elo,

The

average

ower

densitiescarried

by the ncident,

flected,and

ransmitted eams

re

EXERCISE9.T

waveinairisincidentupontheflatbound-

ary

ofa soil

medium

with e.

:

4

and

p,

:

I at9i

:

50'.

Determine1,4, f n,and11. (SeeO)

r r i 1 2

sl=; l ,

(e

t r ; r t2

si

:

r:-!9!.

(9.

-

znl

J,L

:

t F t 1 2

r

"

_L0

...':-'

242



9.5

REFI.ECTIVTTYANDTRANSMISSIVITY

Medium

(er,pr)

Figure 9-18:

Reflection nd ransmission

f an ncident

circularbeam

lluminating spot

of sizeA

on the nter-

face.

where

4y

and

?2

are

he ntrinsic mpedances

f media1

and2, respectively. he

cross-sectional

reasof

the nci-

dent, eflecrcd,

nd ransmitted eams

rc

Tllre eflectivily

R

(also

called

rele

ct4nce

n

optics)

is

defined

as the ratio

of the reflectedpower

to the

inci-

dentpower.

herefl

ectivity

orperpendicularpolarization

is hen

R , = 4 - l E l o l 2 c o s 4

Pl lElnl2os i

: lElo'

l r lo l

(e.76)

l l

where

we

used he

fact that

4

:

8,

in accordance

ith

Snell's

aw

of reflection.

The ratio

of the reflected o

inci-

dent

lectricield

mplitudes,

ElolEiol,

issimply

qual

to he

magnitude

fthe eflection

oefncient

1. Hence,

R1

:

lf1l2,

(9.77)

and,similarly,

or

parallelpolarization

(9.74a)

(9.74b)

(9.74c)

and

hecorresponding

verage

owers

arried

by he

beams

:u€

A i : A c o s d i ,

A, : A cos9r,

A t : A c o s 4 ,

T\e transmissivit!

T

(or

transmittance

in optics) is

de-

fi nedastheratioof

hetransmittedpowertoincidentpower:

R,,

l i

=

;1,,;2.

e.78)

T,

:

PI-

-

lEttol2

lr

-

ptr

lE\olz

nz

A c o s Q

A cosd;

:

lr, l ,

(o'"o.9,')

elea)

\ 4 2 c o s E l

r ,

=

S:

l r ,

' (n ' "o"r ' ) .

(e.7eb)

| i

'

\42coso ; /

ffn"o"e,,

I t i t t2

f,f

o.o,4,

Pi

=

sle,

=EU-

tro"g,.

Pl

:

sler

:

Pi

:

sle.:

(9.7

5a)

(e.75b)

(9.7sc)

The ncident,

eflected.

nd ransmitted

aves o not

have

o obey any

such aws

as conservation

f electric

field,

conservation

f magnetic

ield,

or conservationf

power

density,

but they

do have o

obey he aw of con-

servation

of

power.

n

fact, in many

cases he transmit-

tedelectric

ield s

larger han he

ncident lectric ield.

Conservationfpowerequireshat he ncident ower e



IECHNOL@Y

BRIEF: BAR{ODE.READERS

339

Bar-Gode

eaders

q

bar code

@nsistsof

a sequence

f

parallel

bars

ot

certainwidths,

usually

printed

n black

againsi

a white

bac*ground,

onfigured

o represent

par-

ticular

binary

code of information

bout

a

product

and ts manufacturer.

aser

scanners

an read

he

code

and ranster

he nformation

o

a computer,

:ash register,

r a display

creen.

Forboth

station-

aryscanners

uilt nto

checkout

ounters

t

grocery

stores

ndhandheld

nits

hatcanbe

Dointed t he

oar-coded

biect ike

a

gun,

he basic

operation

f

a

bar-code eader

s the

same

Basic

peration

The

scanner

ses

a laser

beam

of light

pointed

at

a multitaceted

otating

mirror,

spinning

at a high

speedon

the order

of 6,000

evolutions

er

minute

(A).

The rotating

mirror

creates

a

ian

beam

o il-

luminate

he

bar code

on the

obiect.

Moreover,

y

exposing

he aser

ight

o ts

many acets,

t deflects

the

beam ntomany-different

irections,

llowinghe

obiect o be scanndb vera wide angeof positlons

and

orientations.

he

goal

s

to have

one

of those

directions

e such hat

the

beam reflected

y the

bar code

ends

up traveling

n the

direction

f, and

captured

y,

he ight

detector

sensor),

hich

will

read

he

coded

sequence

white

bars reflect aser

light

and

black

onesdo not)

and convert

t into

a bi-

nary

sequence

f

onesandzeros

B).

Toeliminate

interference

y

ambientights,

glass

ilter

s used

as shown

n

(A)to

block

outall ight

except

ora nar

row

wavelength

andcentered

t the wavelength

f

the

aser ight.



340

CHAPTER9

WAVEREFLECTIONAND

TRANSMISSION

PR

Tbble

9-2: Exprcssions

or f, r,

i, md

T for wavc ncidence

rom a mcdium

with intrinsic

impedance

4l

onto a medium

with

intinsic

impodancc

2.

Angles

Q

end

&

rrt tbc angles

of incidenceand

Bnsmission'

espectivcly.

Property

Nonnal

Incid€nce

0 i = f i = Q

Perpendicuhr

Polerization

P.rallel

Polarizption

Reflection

co€fncient

n 1 - n r

4 2 + n l

42cosq - 4t cos0r

?2

os4

+

?l

cos4

F1 42coser

-

4l cos9i

12

cos

Ar

+

lt

cosAi

T[ansmission

coefncient

2rlz

q 2 + 4 \

242cos0i

42

cos

q

+

4l

cos0r

2n2 cos 0l

ttl

?2

cos0t

+

4l

cos0i

Relation of f to

z r : 1 * f

r r : 1 - | f r

r n = ( l + r 1 1 1 {

COS t

Reflectivity

o

-

tc l2

81

:

l r1 l2

R1

=

l f

1 1 2

Transmissivity

r

=

,r,(#)

. n ' c o s A

fr

=

l?,1'

-::--------

42

COS i

. nr

cOS

'

7rr

l rrr l '?

'

4 2 C O S d i

R€lation f R to f

r = l - R 7 r : l - R r

l t = 1 - R i l

Notes:

1)

sin&

=

JplifllEi$sinQ;(2)

41 Jiller:Q)

tt2:

J

pzFal

4)

ornonmagnetic

medta.

2 /n

:

nr /nz

P

Sc

dl l

9.1

of l

les

toll

r r )

,b)

i c

I

CHAPTER

IGHLIGHTS

The

elations escribing

he effection

nd ransmis-

sionbehavior f aplaneEM waveat theboundary

between

wo different

mediaare he

consequencef

satisfyingheconditions

fcontinuity

f the angen-

tial components

f E and

H acrosshe boundary.

Snell's

awsstatehatd1

=

Q

and

o

By successive

multiple

reflections, light can bc

guided

hroughopSical

ibers.The

maximumdatarale

ofdigital

pulses

hatcanbe ransmitted

long ptical

fiberss dictated y

modal ispersion.

At

the Brewsterangle

or a

given polarization, bc

incident

wave s transmittedotally

acrosshebound-

ary. For nonmagnetic

materials, he Brewster

anglc

exists

or

parallel olarization

nly.

Any

planewave

ncident

na

plane

oundary

an

bc

synthesized

s hesum

f aperpendicularly

olarizcd

wave

nda

parallel

olarized

ave.

Transmission-line

quivalent odels an

beused

o

characterize

ave

propagation,

eflection

y,

and

transmission

hroughboundaries

etween

diffa-

entmedia.

6 n

r

tal

. g

sin9r

(nr/nz)

sin0i.

For

media uch hatn2

< nl, the

ncidentwaves re-

flected otally by

the boundary

when

>

9",

where

g. is he ritical ngle iven y8c 5in

l1n2

/n 1 .



?ROBLEMS

341

\

PBOBLEMS

Secllon

-1:Relleclion

ndTnnsmissi0n

rt

lormal

ncldence

9.1*

A

plane

wave n

air with

anelectric

ield

amplitude

of I0 V/m is incidentnormallyupon he surface f a oss-

tess,

onmagnetic

medium with

e,

=

25.

Dctermine

he

-ollowing:

ra) The reflection

nd ransmission

o€fficients.

rb)

The

standing-wave

atio n

theairmedium.

ic)

Theaveragepowerdensities

ftheincident,

eflected,

and

ransmittedwaves.

9.2

A

plane

wave raveling

n medium

with

e.r

:

2.25

r

normally ncident

upon

medium2 with

ea

:

4. 9615

redia aremadeofnonmagnetic,

on-conducting

ateri-

ls. Ifthe electric

ieldofthe

ncident

wave s

given

by

Ei

:

i4cos(62

x l0er

30zr)

(V/m)

a)

Obtain ime-domain

xpressions

or

theelectric

nd

magnetic

ields n each

fthe wo

media.

b) Determine

the

average

ower

densities

of the

inci-

dent,

eflected

nd ransmitted

aves.

t^3

A

plane

wave

raveling

n a medium

with

er,

=

9

.

normally

ncident

upona second

medium

with

e.,

:

-

Bothmedia

aremade

fnonmagnetic,

on-conducting

:aterials.f

themagnetic

ieldof the ncident lane

wave

,

given

by

H\

=

i2 cos(2trx

l}et

-

ky)

(A/m)

a)

Obtain ime-domain

xpressions

or

theelectric

nd

magnetic ields

n each f the

wo media.

b)

Determine

he

average

ower

densities

f the

nci-

dent, eflected,

nd ransmitted

aves.

'Answer(s)

availablen

AppendixD.

I Solutionavailablen CD-ROM.

9.4

A 200.MH4

left-hand

circularly

polarized

plane

wave

with

anelectric

ield

modulus

of 10V/m

is normally

incidcnt

n

air upon

a dielectric

nedium

with

e.

-

4,

and

occupies

he region

defined

by

z

>

0.

(a)

Write

an expression

or

the electric

field phasor

of

the

incident wave,given

that

the field

is a positive

maximum

t

z

=

0andt:0.

(b)

Calculate

he reflection

and transmission

oeffi-

cients.

(c)

Write

expressions

or

heelectric

ieldphasors

f the

reflected

wave,

he transmitted

wave,

and

the total

field

n the egion

3

<

0.

(d)

Determine

he percentages

f the incident

average

power

eflected

y the

boundary

and ransmitted

nto

thesecond

medium.

9.5* Repeat roblem

.4,but

eplace

he

dielecrric

me-

dium

with

a

poor

conductor

haracte

zed

by t

,

:

2 25

&.

=

l,

and

:

l0-4

S/m.

9.6

A 50-MHzplane

wave

with

electric

ield

amplitude

of30 V/m

s normally

ncident

n air

ontoa

semi-infinite,

perfectdielectric

edium

withe.

:

36.Determinethefol-

lowing:

(a)

f

(b)

The

average

ower

densities

of the

incident

and re-

flectedwaves.

(c)

The distance

n

the air

medium

from

the boundary

to

thenearestminimumof

heelectric

ieldintensity,

E .

9.7* What

s

hemaximumamplitudeof

hetotal

electric

fieldin

theair medium

of Problem

9.6,

andat what

nearest

distance

rom

the

boundary

does t

occur?

9.8 Repeat

Problem

9.6, but replace

he dielectric

me-

dium

with

a conductor

with

e,.

:

1,

p. :

1,

and

o :2.78 x l0-r S/m.



342

CHAPTER9

WAVE REFLECTIONAND TRANSMISSION

P

9.9*

The hree egions

hownn Fig. 9-20

contain

perfect

dielectrics.

For a wave n medium

, incident normally

upon

he

boundary

z

=

-d,

what combinationof e.,

andd

produccs

o rcflection?

xpress

our

answersn

lermsofer,

er,and

he

oscillation

requency fthe

wave,

.

J*

d--------1

z =

d

z = 0

Figure 9-20:

Dielecric

layers or Problems9.9

to

9.1l.

9.10

For the configuration

shown in Fig. 9-20, use

transmissionline

equations

(or

the

Smith

chart) to

calculate

he input impedanceat

z

:

-d

for et,

:

1,

e,,

:

9,

e,,

=

4, d

--

1.2m, and

:

50 MHz. Also de-

termine the fraction of

the incident average

power

density

reflectedby the structure.Assume

all media are ossless

andnonmagnetic.

9.11.

RepeatProblem9.10,but

nterchange r, and err.

9.12

Orange

ight of

wavelength

.61

pm

in air enters

a block of

glass

with

ot

=

2.25. What color would it

appear

o a sensor mbeddedn the

glass?The

wavelength

rangesof colorsare violet (0.39 o 0.45 plm), blue (0.45

to 0.49

pm), green 0.49

o

0.58

prm),yellow

(0.58

o

0.60

pm),

orange

0.60

o 0.62

pm),

and ed

(0.62

s

0.78

pm).

9.13*

A

plane

wave

of unknown requency s normally

incident n air upon the surfaceof a perfectconductor.

Using an electric-fieldmeter,

t was determined

hat

the

total electric

ield in the air medium s always

zero

when

measured t a

distance

f

2.5 m from the conductor

ur-

face.Moreover, o suchnullswereobserved t distances

closer o the conductor.

What s

the requency f

the

n-

cidcnt

wave?

I

9,1,1

Consider

thin ilm of soapn air under

llumina-

tion by

yellow

ight with I

:

0.6

pm

in vacuum.

f

thc

film is treated sa

planar

ielectric lab

with

e.

:

1.72,

surrounded n both sidesby air,

what film

thickness

would

produce

trong eflection f the

yellow

ight

at

normal ncidence?

9.15-

A 5-MHz

plane

wavewith electric ield ampli-

tude

of

20

(V/m)

is normally ncident n air

onto he

plane

urface f a semi-infiniteonductingmaterial

widl

€,

:

4,

p,:t

l. ando

:

100

S/m).

Determinehe

average

ower

dissipated

lost)

per

unit cross-sectional

area n a 2-mm

penetration

f theconductingmedium.

9.16

A

0.5-MHz

antenna aried

by anairplane

lying

over he ocean urface

generates

wave

hat approaches

the

water

surface

n

the orm of

a normally

ncident

plane

wavewith

an

electric-field mplitude

f 3,000

V/m).

Seawaterscharacterizedy e'

:

72.

L:

l,ando

=4

(S/m).

The

plane

s trying to communicate message

to a submarineubmerged

t a

depthd below

he

water

surface.

f the

submarine's

eceiver

equires

minimum

signal mplitudef 0.1

(pVlm).

what

s the

maximum

depthd

to whichsuccessfulommunications still

pos-

sible?

g

9

tt

fit

s

d.

C

n

7

rl

P



PROBLEMS

343

Figure

9-21:

Prism

of

hoblem

9.17.

S8clions

-2ald

9-3:Snell's

aws ndFibel

plics

9.17-

A

light ray

is incident

on

a

prism

n

air at

an

angle

asshown

n Fig.

9-21.The

ay s

refracted

t he

first

surfaceand

againat

the second

urface.

n

terms

of

the

apex

angle

@

f the

prism

and ts ndex

ofrefraction

n,

determinehe smallestvalueof d for which theray will

emerge

rom the

other

side. Find

this

minimum

d

for

n :

1 .5 ndd

=

60 ' .

9.18

For

some

types of

glass,

he

index

of refraction

laries

with

wavelength.

A

prism

madeof

a material

with

n

:

l .7 l

-

: :

lo ,

J U

.r'here

1.6

s the wavelength

n vacuum,

was

used

o dis-

:ersewhite

ight

asshownn

Fig.9-22.

Thewhite

ight

s

Figure

9-23:

Periscoperisms

of

problem

9.19.

incident

at

an angle

of

50" the

wavelength

,s

of red ight

is 0.7

pm,

and

rhat

of violet

light

is

0.4

pm.

Determine

the

angular

dispenion

n degrees.

9.19-

The

two prisms

n Fig.

9-23 are

made

of

elass

with

z

:

1.

52.What

raction

f

the

power

ensity

airied

by theray incidentupon he top prismemergesrom the

bottomprism?

Neglect

multiple

ntemal

eflections.

9.20 A light ray ncident

t 45.

passes

hrough

wo

die_

lectric

materials

ith

the

ndices

f

refraction

nd hick_

nessesiven

n Fi

9.9-24.If

the ay

strikes

he

surface

f

the irst

dielectric

t a height

of

2 cm,

at what

heieht

will

it

strike he

screen?

9.21*

Figure

9-25depicts

beaker

ontaining

block

of

glass

n

thebottom

and

water

over

t. The

slassblock

contains

smallair

bubble

r

an unknown

Jnth

below

the water

surface.

When

viewed

rom

above

i an ansle

of 60",

he

air bubble

ppears

t a depth

f

6.g

cm.

Wf,ar

is the ruedepthof theairbubble?

().6

n

pm)

Figure

9-22:

Prism

f Problem

.18.



344

CHAPTER

9 WAVE

REFLECTION

AND

TRANSMISSION

P

T

2rm

t

L-l-r-.

'

3cm 4cm 5cm

Figure9-Z:

Light ncident n a scre€nhrough multi-

layered ielectric

Problem

.20).

-f

|

6 .8 lcm

Apparent osition

of air

bubble

Air bubble

Figure 9-25:

Apparent

position

ofthe air bubble

n Prob-

lem 9.21.

9.22

A

glass

emicylinder ith

n

:

1.5 s

positioned

such hat ts lat ace

s horizontal, s

shown

n Fig.

9-26,

and ts

horizontal urface upports drop

of oil, asalso

shown.When ieht is directed adiallv oward he oil.

total intemal

reflection

occurs f

g

exceeds

0".

What

s

the ndex

of refraction

of the oil?

Figure 9-26:

Oil drop

on the flat

surfaceof

a

glass

semi-

cylinder

(Problem

9.22).

9.23-

A,

penny

lies at the

bottom

of a water

fountain

at a

depth of

30 cm. Determine

he

diameter

of

a

piece

of paper

which,

if

placed

to ffoat

on the

surface

of the

water

directly

above

he

penny,

would

totally

obscure he

penny

from view. Treat

the

penny

as

a

point

and assume

that

n

:

1.33

or water.

9.24

Suppose

hat the optical

fiber

of Example

9-5

is

submergedn water (with n : 1.33) nsteadof air. De-

termine

0a and

/p

in that

case.

9.25-

Equation

(9.45)

was

derived

for

the casewhere

the light

incident

upon the

sending

end of

the optical

fiber

extends

over the

entire acceptance

one shown

n

Fig. 9- 2(

b ). Suppos,5:

he incident

ight

is consrrainedo

a

narrower range

ex'tending

between

normal

incidence

and

0'

,

where

0'

<

0^-

(a)

Obtain an

expression

or the maximum

data ate

p

in

terms of 8'.

(b)

Evaluate

/p

for

the fiber

of Example

9-5

when

e ' 3 . .

$

a

ls

(

(

(

l l

(

'g

r

9

6

I

IE

L

P

it

F

I

6

i

electromagnetic 9

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