field and wave electromagneticocw.snu.ac.kr/sites/default/files/note/1001.pdf · 2018-01-30 ·...

Field and Wave Electromagnetic

Chapter8

Plane Electromagnetic Waves

Electromagnetic Theory 2 2

Introduction (1)

Homogeneous vector wave equation

Time harmonic, unbounded homogeneous medium· Parameter : intrinsic impedance, attenuation constant, phase constant· Skin depth : the depth of wave penetration into a good conductor· Poynting vector : power flux density· plane wave incident normally on a plane boundary· reflection and refraction on a plane boundary· no reflection and total reflection

22

2

7 90 0

8

0 0

0

14 10 , 10

361

3 10

EE

t

c

με

μ μ π ε επ

μ ε

− −

∂∇ − =

∂

= = × = = ×

= ≅ × ⇒

In free space

Let m/s velocity of wave propagation in free space


Introduction (2)

Uniform plane wave: Assuming the same direction, magnitude, phase of fields in infinite

planes the direction of propagation ⊥


Plane Wave in Lossless Media (1)

Homogeneous vector Helmholtz’s equation (assuming time harmonic )

In cartesian coordinates

Uniform plane wave→uniform Ex over plane surface z

j te ω

2 20 0 0 0 00, where [rad/m]E k E k

c

ωω μ ε∇ + = = =

2 2 22 2 2

0 02 2 20 0xE k E k E

x y z

⎛ ⎞∂ ∂ ∂∇ + = ⇒ + + + =⎜ ⎟∂ ∂ ∂⎝ ⎠

⊥

0 0

2 2

2 2

22

2

0 0

0 0

. ) 0, 0

0

( ) ( ) ( )

x x

xo x

jk z jk zx x x

E Ei e

x y

Ek E

z

E z E z E z E e E e

E E

− ++ − + −

+ −

∂ ∂= =

∂ ∂

∂∴ + =

∂∴ = + = +

: ordinary differential equation

where and : arbitrary constant satisfying boundary condition



Assume cos(ωt), : real (zero reference phase at z =0)

Phase velocity : the velocity of propagationof an equiphase front

wavenumber

0E+

0( )0 0 0( , ) Re[ ( ) ] Re[ ] cos( )j t k zj t

x xE z t E z e E e E t k zωω ω−+ + + += = = −

0 : constant phaset k z Aω − =

0 0 0

1

: the velocity of propagation of an equiphase front=the velocity of light

p

dzu c

dt k

ωμ ε

= = = =

00

2 2 : number of wavelength in a complete circle

fk

c c

ω π πλ

= = =



wave length : 00

2

k

πλ =

00 : traveling wave in the -z directionjk zE e−

0

0

0

0 0

0 00

0 0 0

00

0

0 0 ( )

( ) 0 0

( )10, , 0

( )( ) ( )

1( ) ( ) ( )

120 377

x y z

x

xx y z

jk zxx

y x x

x y z

E j xH yH zHz

E z

E zH H H

j z

E zE e jk E z

z z

kH z E z E z

k

ωμ

ωμ

ωμηωμ η

μη πε

+ + +

+

++ + +

+−+ +

+ + +

∂∇× = = − + +

∂

∂∴ = = − =

∂

∂ ∂= = −

∂ ∂

∴ = = ⇒ =

∴ = ≅ = Ω

: intrinsic impedance of the

00

0

( , ) ( , ) Re ( ) cos( )j ty y

EH z t yH z t y H z e y t k zω ω

η

++ +⎡ ⎤= = = −⎣ ⎦

free space


Doppler Effect (1)

Assume transmitter moves at the velocity of Wave radiated at t=0 arrives at the Rx at t=t1.Wave radiated at t= t arrives at the Rx at t=t2.

the time elapsed at Rx, corresponding to t at Tx is

u

01

rt

c=

1/ 22 22 0 0

2 2 00 2

0

1/ 2

' 12 cos ( )

( ) , 1 cos

1) (1 ) 1

2

rt t t r r u t u t

c c

r u tu t r t t

c r

cf x x

θ

θ

⎡ ⎤= Δ + = Δ + − Δ + Δ⎣ ⎦

⎛ ⎞ΔΔ Δ + −⎜ ⎟

⎝ ⎠

− −

if

2 1' 1 cosu

t t t tc

θ⎛ ⎞Δ = − = Δ −⎜ ⎟⎝ ⎠

u


Doppler Effect (2)

Assume t is period of the time harmonic source

then

1. ) i e t

fΔ =

21

' 1 cos 1' 1 cos

f u u

f fut c cc

θθ

⎛ ⎞ ⎛ ⎞= = +⎜ ⎟ ⎜ ⎟Δ ⎝ ⎠ ⎝ ⎠−∵


TEM Wave (1)

Uniform plane wave characterized by propagating alongz axis → particular case of a transverse electromagnetic wave

General form of TEM wave

satisfies the homogeneous Helmholtz’s equation, provided that

, x yE xE H yH= =

0 0( , , ) yx zjk yjk x jk zj k rE x y z E e E e e e−− −−= =i

2 2 2 2x y zk k k ω με+ + =

x y zk xk yk zk kn

r xx yy zz

⎧ = + + =⎪⎨

= + +⎪⎩

Wave number vector :

Position vector from the origin :

when constant is a plane of constant phase and uniform amplitude for the wave.

n r =i


TEM Wave (2)

0 0 0

0

0

( )

0

( ) ( ) ( ) 0

( ) 0

( ) ( ) (x y z

j k r j k r j k r

j k r

j k x k y k zj k rx

E

E E e E e E e

E

E e

e x y z e j xk ykx y z

− − −

−

− + +−

∇ =

∇ = ∇ = ∇ + ∇ =

→

∴ ∇ =

⎛ ⎞∂ ∂ ∂∇ = + + = − +⎜ ⎟∂ ∂ ∂⎝ ⎠

i i i

i

i

i

i i i i i

∵

i

(Source free)

plane wave is a constant vector

( )

0

0 0

)

( ) 0

0

x y zj k x k y k zy z

j k r

j k r

zk e

jke

j E k e

k E E

− + +

−

−

+

= −

∴− =

∴ = ⇒

i

ii

i

is transverse to the direction of propagation


TEM Wave (3)

Remark : A uniform plane wave propagating in an arbitrary direction is a TEM wave with are normal to

0

0 0 0

1 1( ) ( ) ( ) ( )

1( ) ( ),

1( ) ( )

( )

) ( )

j k r

j k r j k r j k r

H r E r jk E rj j

kn E r n E r

k

H r n E e

A A A

cf E e e E e E

ωμ ωμ

ωμ μηωμ η ε

η

ψ ψ ψ

−

− − −

= − ∇× = − − ×

= × = × = =

∴ = ×

∇× = ∇× +∇ ×

∴∇× = ∇× +∇ ×

i

i i i

where

0 ( )j k rjk E e jkn E r−= − × = − ×i

n and and E H E H⊥ n


Polarization of Plane Waves (1)

Linearly polarizedif) vector of the plane wave is fixed in the x direction

Superposition of two linearly polarized waves→One in the x-direction. The other in the y-direction and lagging in

time phase

set z=0,

E

. ) xi e E xE=

1 2 10 20

1 2

10 20

( ) ( ) ( )

( , ) Re{ ( ) ( ) }

cos( ) cos2

jkz jkz

j t

E z xE z yE z xE e y jE e

E z t xE z yE z e

xE t kz yE t kz

ω

πω ω

− −= + = −

⎡ ⎤= +⎣ ⎦⎛ ⎞= − + − −⎜ ⎟⎝ ⎠

( j implys 90 lagging in time)

1 2 10 20(0, ) (0, ) (0, ) cos sinE t xE t yE t xE t yE tω ω= + = +

90



the tip of the vector → traversing an elliptical locus in the counter clockwise direction

(0, )E t

2

210 2 1

10 20 10

2 2

2 1

20 10

(0, ) (0, ) (0, )cos , sin 1 cos 1

(0, ) (0, )1

E t E t E t

t t tE E E

E t E t

E E

ω ω ω⎡ ⎤

= = = − = − ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤∴ + =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

10 20

10 20

if : elliptically polarized

if : circularly polarized

E E

E E

≠=

10 20

1 2

1

,

(0, )tan

(0, )

When the instantaneous angle

E E

E tt

E t

α

α ω−

=

= =



right hand polarized.if is lagging 90° in time phaseleft hand polarizedif is leading 90° in time phase

Linearly polarizedif and are in space quadrature , but in time phase

2E

2E

10 20

10 20

1 2

1

)

( )

(0, ) cos sin

(0, )tan

(0, )

time lagging

jkz jkz

cf

E z xE e y jE e

E t xE t yE t

E tt

E t

ω ω

α ω

− −

−

= +

= −

= = −

2 ( )E z 1( )E z

10 20(0, ) ( ) cosE t xE yE tω= +


Plane Wave in Lossy Media (1)

In a source-free lossy medium

Conventional notation used in transmission-line theory

For a lossless medium,

2 2 0, where c c cE k E k ω με∇ + = =

1/ 2 1/ 2''

1 ' 1'

) ' ''

propagation constant

c c

c

jk j

j j j jj

cf j

γ ω με

σ εγ α β ω με ω μεωε ε

ε ε ε

= =

⎛ ⎞ ⎛ ⎞= + = + = −⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

= −

0 ( '' 0, ')σ ε ε ε= = =

0, kα β ω με= = =

2 2 0Then E Eγ∇ − =


Solution → Uniform plane wave propagating in the +z direction

Low loss dielectrics: means a good but imperfect insulator,

0

0

)1 20log 8.69

zx

z j zx

E xE xE e E

cf Np e dB

E E e e

γ

α β αβ

−

− −

= =

= =

⎧= ⇒ ⎨

⎩

assuming that is linearly polarized in the x direction

: attenuation constant (Np/m)Then

: phase constant (rad/m)

'' ' 1 or σε εωε

2

2

'' 1 ''' 1

2 ' 8 '

'' 1 ''' 1

2 ' 8 '

) ' , ''

then

(Np/m),

j j j

cf

ε εγ α β ω μεε ε

ωε μ εα β ω μεε ε

σε ε εω

⎡ ⎤⎛ ⎞= + − +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞∴ ≅ ≅ +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

= =



Intrinsic impedance

phase velocity

1/2

( )

'' ''1 1 :

' ' ' 2 '

,

,

xc

y

j

j t j t

Ej j E H

H

HE H E Ee

E e H e

φ

ω ω φ

μ ε μ εηε ε ε ε

η ηη

−

+

⎛ ⎞ ⎛ ⎞= − + ⇒⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= = =

∼ ∼

complex number ( and are not in time phase)

: lagging in time phase,

21 1 ''

18 ''

(m/s)puω εβ εμε

⎡ ⎤⎛ ⎞= ≅ −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦



Good conductors1

σωε

1/ 2

1/ 22 4

11

2

1) ( )

2

(1 ) , ,

j j

jj j j

j j

jcf j e e

j j f f

π π

σ σγ ω με ω με ωμσ ωμσωε ωε

γ α β π μσ α β σ

⎛ ⎞ += + = =⎜ ⎟

⎝ ⎠+

= = =

∴ = + + = ∝

fα β π μσ= =

4

(1 ) (1 )

,

[ ]

magnetic field lags behind the electric field intensity by 45

cc

j

c c

j fj j

j

eπ

μ μ ωμ π μ αη σε σ σ σω

η η

= ≅ = = + = + Ω−

= °



phase velocity

wavelength

2pu

ω ωβ μσ

=

7 7. ) 5.8 10 4 10

720

copper [s/m], [H/m] [m/s] at 3 MHzp

e g

u

σ μ π −= × = ×=

22 ,pu

f f

π πλβ μσ

= = =

6 7 7 4

4

0.24 100( )

(3 10 )(4 10 )(5.8 10 ) 2.64 10

1 110 0.038 38 0.66

2.62

1 1 1

2

3MHz ( ), in the air

(Np/m)

at 3MHz at 10GHz

: skin depth or for good conductor

mm m

m mm m m

f

λ λ

α π π

δ μ μα

λδ δ α βα β ππ μσ

−

−

→ = =

= × × × = ×

= = × = = =

= = = = =∵



Note ( )( ) , :

( , )( , )

complex value

cf) mistake

xy c

c

xy

c

E zH z

E z tH z t

ηη

η

=

= ⇒



Group Velocity (1)

lossless medium is a linear function of ω

Information-bearing signals : a band of frequencies → waves of the component frequencies travel with different phase velocities → distortion in the signal wave shape → signal disperse →dispersion → lossy dielectric is a dispersive medium.

i.e) Information-bearing signal → small spread of frequencies around a high carrier frequency → group of frequencies → wave packet

Phase velocity (m/s)puωβ

=

β ω με=1

: independent of frequencypuμε

=


Group Velocity (2)

Group velocity : the velocity of propagation of the wave-packet envelope ( a group of frequencies )

Simplest case : wave packet consisting of two traveling waves having equal amplitude and slightly different frequencies phase constants

0 0, ω ω ω ω+ Δ −Δ

∴

,β β β β+ Δ −Δ

( ) ( ) ( ) ( )( ) ( )

0 0 0 0 0 0

0 0 0 0

( , ) cos cos

2 cos cos ,

E z t E t z E t z

E t z t z

ω ω β β ω ω β β

ω β ω β ω ω

⎡ ⎤ ⎡ ⎤= + Δ − + Δ + −Δ − −Δ⎣ ⎦ ⎣ ⎦= Δ − Δ − Δ


Group Velocity (3)

Phase velocity

the velocity of envelope ( group velocity, )

00 0

0

constant, p

dzt z u

dt

ωω ββ

− = = =

gu

1

/

10,

/

g

g

dzt z u

dt

ud d

ωω ββ β ω

ωβ ω

ΔΔ − Δ = = = =

Δ Δ Δ

Δ → =

constant,

2

1, ,

1

pp g

p p p

pg

p

p

dud d du u

d d d u u u d

uu

du

u d

ω ω β ω ωβ β ω ω ω

ωω

⎛ ⎞= = = = −⎜ ⎟⎜ ⎟

⎝ ⎠

=−


Group Velocity (4)

No dispersion

Normal dispersion

Anomalous dispersion

0 ( is independent of , is a linear function of )pp

duu

dω β ω

ω=

g pu u∴ =

0, ( decreasing with )pp

g p

duu

du u

ωω

<

<

0 ( increasing with )pp

g p

duu

du u

ωω

>

>


Poynting Vector (1)

Vector identity

B

Et

DH J

t

⎡ ∂∇× = −⎢ ∂⎢

⎢ ∂∇× = +⎢ ∂⎣

( ) 2

2

2 2

( ) ( ) ( )

1 1

2 2

1

2

1 1( )

2 2

B DE H H E E H H E E J

t t

B HH H H H H

t t t t

DE E

t t

E H H Et

μ μ μ

ε

μ ε

∂ ∂∇ × = ∇× − ∇× = − − −

∂ ∂∂ ∂ ∂ ∂ ⎛ ⎞= = = ⎜ ⎟∂ ∂ ∂ ∂ ⎝ ⎠

∂ ∂ ⎛ ⎞= ⎜ ⎟∂ ∂ ⎝ ⎠∂ ⎛∴∇ × = − +∂ ⎝

i i i i i i

i i i

i

i

where

2Eσ⎞ −⎜ ⎟⎠


Poynting Vector (2)

2 2 21 1( )

2 2s V VE H ds E H dv E dv

tε μ σ∂ ⎛ ⎞× = − + −⎜ ⎟∂ ⎝ ⎠∫ ∫ ∫i

2 2

2 2

1 1

2 2

1 1

2 2

V

E H

E H dvt

E H

ε μ

ε μ

×

∂ ⎛ ⎞− +⎜ ⎟∂ ⎝ ⎠

+

∫

: power flow per unit area

: the rate of decrease of the electric and magnetic energies stored

where, : energy stored in electric and magnetic

2

VE dvσ∫

field

: Ohmic power dissipation or heat

) :

S

P E H

P ds

= ×

∫ i

2 (W/m Poynting vector

: the power leaving the enclosed volume (Poynting theorem)


Poynting Vector (3)

Instantaneous and average power densities.

Assuming phasor ( )0( ) ( ) j j z

xE z xE z xE e α β− += =

* *2 2

*2*2

( )

1 1 1 1,

2 2 2 2

e mS V V

e m

P ds w w dv p dvt

w E E E w H H H

J J Jp E E E

σ

σ

ε ε μ μ

σ σσ σ

∂− = + +

∂

= = = =

= = = =

∫ ∫ ∫i

i i

ii

or

( )0

( )0

0

( , ) Re ( ) Re

( ) ( ) ,

( , ) Re ( )

Then

propagating in a lossy medium in the +z direction

where

j t z j t z

j z jzy

j t

E z t E z e xE e e

EH z yH z y e e e

EH z t H z e y e

η η

ω α ω β

β θ θα

ω

η ηη

η

− −

− +−

−

⎡ ⎤ ⎡ ⎤= = ⎣ ⎦⎣ ⎦

= = =

⎡ ⎤∴ = =⎣ ⎦ cos( )z t zαηω β θ− −


Poynting Vector (4)

* * * * * *

( , ) ( , ) ( , )

Re ( ) Re ( ) Re ( ) ( )

1 1 1) Re( ) Re( ) ( ) ( ) ( ) ( )

2 2 4

j t j t j t

P z t E z t H z t

E z e H z e E z H z e

cf A B A A B B A B A B A B A B

ω ω ω

= ×

⎡ ⎤ ⎡ ⎤ ⎡ ⎤× ≠ ×⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤× = + × + = × + × + × + ×⎢ ⎥⎣ ⎦

But,

*1

Re2

A B A B⎡ ⎤= × + ×⎢ ⎥⎣ ⎦

220

220

220

0

( , ) ( , ) ( , ) cos( ) cos( )

cos cos(2 2 )

1( ) ( , ) cos

2

z

z

T zav

EP z t E z t H z t z e t z t z

Ez e t z

EP z P z t dt z e

T

αη

αη η

αη

ω β ω β θη

θ ω β θη

θη

−

−

−

∴ = × = − − −

⎡ ⎤= + − −⎣ ⎦

= =∫ 2

(W/m )


Poynting Vector (5)

or * 2

* *

1( , ) Re ( ) Re ( ) Re ( ) ( ) ( ) ( )

21 1

( ) Re ( ) ( ) ( ) Re2 2

j t j t j t

av av

P z t E z e H z e E z H z E z H z e

P z E z H z P z E H

ω ω ω⎡ ⎤⎡ ⎤ ⎡ ⎤= × = × + ×⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤∴ = × ⇒ = ×⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦2General form (W/m )


Normal Incidence at a Plane Conducting Boundary (1)

1 1

1

( ) , ( )

( ) ( ) ( )

j z j zioi iio

i i i

EE z xE e H z y e

P z E z H z

β β

η− −= =

= ×

In medium 1,

2 20, 0In medium 2, E H= =

Medium 1 Medium 2

1

1 1

0

1 0 0

1 2 1 2 0 0

1

( )

( ) ( ) ( ) ( )

(0) (0) 0 ),

( )

j zr r

j z j zi r i r

t t i r

E z xE e

E

E z E z E z x E e E e

E E E E E E

E z xE

β

β β−

∴

=

∴

= + = +

= = = ∴ = −

=∴

Incident wave is reflected on the boundary

Total field in the medium1

(

1 10

0 1

( )

2 sin

j z j zi

i

e e

x j E z

β β

β

− −

= −


Normal Incident at a Plane Conducting Boundary (2)

1 1( ) ( )No average power since and are in phase quadratureE z H z

1 1

1 1

00

1 1

01 1

1

1 1( ) ( ) ( ) ( )

1( )

( ) ( ) ( ) 2 cos

r r r

j z j zir r

ii r

H z n E z z E z

EH z y E e y e

EH z H z H z y z

β β

η η

η η

βη

= × = − ×

= − =

∴ = + =

* *1Re( ) 0,

2avP E H E H= × = ×∵ cf) : imaginary



Space time behavior of the total field

1 0 1

01 1

1

( , ) 2 sin sin

( , ) 2 cos cos

standing wave

i

i

E z t x E z t

EH z t y z t

β ω

β ωη

⎡ =⎢ ⇒⎢ =⎢⎣

( ) ( )

1

11

1

11

( , )

2( , )

( , )2 1 2 1

2 4( , )

zeros of occur at or

Maxima of

Maxima of occur at or

zeros of

E z tz n z n

H z t

E z tz n z n

H z t

λβ π

π λβ

⎫⎪ = − = −⎬⎪⎭

⎫⎪ = − + = − +⎬⎪⎭



1 0 0

1

1

00 0

1

r i

ir i

E E E

E

H

EH H

λ

η

⇒ = −

= = ⇒

Note

vanishes on the conducting boundary

vanishes at points of multiples of from the boundary2

is maximum on the conducting boundary

Bound

2

1 1

) .331 ( ( ) )s scf P n H J n H H J

E H

× = × − =

°

1 12 2

ary condition

,

The standing waves of and are in time quadrature (90 phase difference) and are shifted in space by a quarter wave length


Oblique Incidence at a Plane Conducting Boundary (1)

Depending on the polarization of incident wavePlane of incidence : defined by the normal vector to the boundary

and the wave vector: orientation of the incident field

iE can be decomposed into two components : one perpendicular to, and the other parallel to the plane of incidence

Perpendicular polarization

sin cos

: angle of incidencei i i

i

n x zθ θθ

= +

iE E



For the reflected wave, sin cos : angle of reflection

r r r

r

n x zθ θθ

⎧ = −⎪⎨⎪⎩

11 ( sin cos )0 0

( )0 1 1

( )0

1 1

( , )

sin , cos

1( , ) ( , ) ( cos sin )

i i i

x z

x z

j x zj n Ri i i

j x zi x i z i

j x zii ii i i

E x z yE e yE e

yE e

EH x z n E x z x z e

β θ θβ

β β

β β

β β θ β β θ

θ θη η

− +−

− +

− +

= =

= = =

⎡ ⎤= × = − +⎣ ⎦

i

where

1

1 1

( sin cos )0

sin sin0 0

0 0 1 1

( , )

( ,0) ( ,0) ( ,0) 0

sin sin

at z=0,

hold for all value of x and

: Snell's law of reflection

r r

i r

j x zr r

j x j xi r i r

i r i r

r i

E x z yE e

E x E x E x y E e E e

E E x x

β θ θ

β θ β θ

β θ β θθ θ

− −

− −

=

⎡ ⎤= + = + =⎣ ⎦⇒

= − =∴ =



1

1

( sin cos )0

( sin cos )0

1 1

( , )

1( , ) ( , ) ( cos sin )

i i

i i

j x zr i

j x zir rr i i

i r

i r

E x z yE e

EH x z n E x z x z e

x H H

z H H

β θ θ

β θ θθ θη η

− −

− −

= −

⎡ ⎤= × = − −⎣ ⎦

component of and are in the same direction

component of and

⎡⎢⎢⎣ are in the opposite direction

1 1 1 1

1

1

cos cos sin sin0 0 1

sin01 1 1

1

( , ) ( , ) ( , )

( ) 2 sin( cos )

( , ) 2 cos cos( cos ) sin sin( cos

i i i i

i

i r

j z j j x j xi i i

j xii i i i

E x z E x z E x z

yE e e e y j E z e

EH x z x z e z j z

β θ β θ β θ β θ

β θ

β θ

θ β θ θ β θη

− − −

−

= +

= − = −

= − +

The total field

1 sin) ij xe β θ−⎡ ⎤⎣ ⎦



1 1

1 1

1 1

sin cos

cos

Note In the direction normal to the boundary and

standing wave patterns according to and where

No average power is propagated

In the direction

y x

z z

z i

E H

z zβ ββ β θ

→

⇒= →

1 1

1 11 1

1 1 1

2,

sin sin sin

parallel to the boundary and are in both time and

space phase propagate with a phase velocity

The propagating wav

y z

x xx i i x i

E H

uu

λω ω πλβ β θ θ β θ

→

→

= = = = =

1 1 11

1

20 sin( cos ) 0 cos cos , 1,2,3...

2cos

e in the x direction is a nonuniform plane wave

for all x when or

a conducting plate could be inserted at without changi

i i i

i

E z z z m m

mz

πβ θ β θ θ πλ

λθ

= = = = − =

= −

1 0

ng the field

pattern that exists between the conducting plate and the conducting boundary at z=0 TE wave( ) would bounce back and force between the conducting

planes and xE⇒ =

propagate in the x direction



and have x- and y- component

and have only y- component

i r

i r

E E

H H

1

1

( sin cos )0

( sin cos )0

1

( , ) ( cos sin )

( , )

i i

i i

j x zi i i i

j x zii

E x z E x z e

EH x z y e

β θ θ

β θ θ

θ θ

η

− +

− +

= −

=

Parallel polarization



The reflected wave

1

1

( sin cos )0

( sin cos )0

1

( , ) ( cos sin )

( , )

r r

r r

j x zr r r r

j x zrr

E x z E x z e

EH x z y e

β θ θ

β θ θ

θ θ

η

− −

− −

= +

= −

0At z=0, for all xtE =

1 1

1 1 1 1 1 1

sin sin0 0

0 0

1

cos cos sin cos cos sin0 0

0

( cos ) ( cos ) 0

( , ) ( , ) ( , )

cos ( ) sin ( )

2

i r

i i i i i i

j x j xi i r r

r i i r

i r

j z j z j x j z j z j xi i i i

i

E e E e

E E

E x z E x z E x z

xE e e e zE e e e

E

β θ β θ

β θ β θ β θ β θ β θ β θ

θ θθ θ

θ θ

− −

− − − −

∴ + =

∴ = − =

∴ = +

= − − +

= −

and

1

1

sin1 1

sin01 1

1

cos sin( cos ) sin cos( cos )

( , ) ( , ) ( , ) 2 cos( cos )

i

i

j xi i i i

j xii r i

x j z z z e

EH x z H x z H x z y z e

β θ

β θ

θ β θ θ β θ

β θη

−

−

⎡ ⎤+⎣ ⎦

= + =


Normal Incidence at a Plane Dielectric Boundary (1)

Assume lossless (σ1=0, σ2=0) media

Incident wave travels in +z directon

1 100

1

( ) , ( ) j z j zii ii

EE z xE e H z y eβ β

η− −= =



The boundary surface is the plane z=0.→ Discontinuity at z=0 cause that the incident wave is partly reflected back and partly transmitted into medium 2Reflected wave ( )

4 Transmitted wave ( )

We have two unknown thus two equations are required todetermine . Boundary conditions for the electric and magnetic fields

,r rE H

1 100

1 1

1( ) , ( ) ( ) j z j zr

r r rr

EE z xE e H z z E z y eβ β

η η= = − × = −

,t tE H

2 200

2 2

1( ) , ( ) ( ) j z j zt

t t tt

EE z xE e H z z E z y eβ β

η η− −= = × =

0 0,r tE E

0 0,r tE E



can be positive or negative depending on is always positive→ are in same direction

Γ

0 0 0

00 0

1 2

(0) (0) (0),

1(0) (0) (0), ( ) no current

i r t i r t

ti r t i r

E E E E E E

EH H H E E

η η

+ = + =

+ = − = ∵

2 1 2 10 0

2 1 2 1

2 20 0

2 1 2 1

0 02 1 2

0 2 1 0 2 1

, : Reflection coefficient

2 2 , : Transmission coefficient(Dimensionless)

2 ,

r i

t i

r t

i i

E E

E E

E E

E E

η η η ηη η η η

η ηη η η η

η η ητη η η η

− −∴ =

+ +

=+ +

−Γ = = = =

+ +

2η

τ and i tE E



The definition for apply even when the media are dissipativei.e) and/or are complex

Relation between reflection and transmission coefficient

and τΓ1η 2η

2 1

2 2* 2 00 02 2

0 2 22 1 2 1 1 12 1 2 1

2*

0 00

1 1

) 1 if

i.e

But

422 4 1 ( 0)

t i

t t i i

t i

ii it t i

i ii i i

cf

E EE H E H

H H

EE EE H E

E EE H E

τ η η

ηη ηη η η η η ηη η η η

η η

> >

⎫> ⎪⇒ × < ×⎬< ⎪⎭

× = × = < − <+ + + +

× = × =

∵

1 τ+ Γ =



if (i.e medium 2 is perfect conductor)

if , partial reflection

Two parts : A traveling wave with an amplitudeA standing wave with an amplitude

never goes to zero at fixed distance from the interface but has merely locations of minima and maxima

2 0η =

1, 0τΓ = − =

2 0η ≠

1 1

1 1 1

1

1

1 0

0

0 1

1 0 1

( ) ( ) ( ) ( )

(1 ) ( )

(1 ) ( 2sin )

or ( ) ( 2sin )

j z j zi r i

j z j z j zi

j zi

j zi

E z E z E z xE e e

xE e e e

xE e j z

E z xE e j z

β β

β β β

β

β

β

τ β

−

− −

−

−

= + = + Γ

⎡ ⎤= + Γ +Γ −⎣ ⎦

⎡ ⎤= + Γ +Γ⎣ ⎦

⎡ ⎤= + Γ⎣ ⎦

0iEτ

02 iEΓ1 ( )E z∴



Location of minima and maxima.Assume lossless

1 121 0( ) (1 )j z j z

iE z xE e eβ β−= + Γ

2 1

1 0

1 max

1max

1

1 0

case 1, 0 ( )

the maximum value of ( ) is (1 )

which occurs when 2 2 ( 0)

, 0,1, 2,...2

the minimum value of ( ) is (1 )

i

i

E z E

z n z

nnz n

E z E

η η

β πλπ

β

Γ > >

+Γ

⇒ = − <

∴ = − = − =

−Γ

∵

1 min

1min

1

which occurs when 2 (2 1)

(2 1)(2 1) , 0,1, 2,...

2 4

z n

nnz n

β πλπ

β

⇒ = − +++

∴ = − = − =



standing wave ratio(SWR), s→expressed in dB scale (20log10 s)

An inverse relation

max

min

1, 1

1

Es s

E

+ Γ= = < < ∞

− Γ

2 1

1 0

max

1 0

min

case 2, 0 ( )

the maximum value of ( ) is (1 )

which occurs at (2 1)

the minimum value of ( ) is (1 )

which occurs at 2

i

i

E z E

z n

E z E

z n

η η

π

π

Γ < <

−Γ

⇒ = − +

+Γ

⇒ = −

1 maxThe location for ( ) when 0 and when 0

are interchanged

E z⇒ Γ > Γ <

1, 1 1

1

s

s

−Γ = − < Γ <

+



Magnetic field intensity

In lossless medium, is real; will be a minimum at locations where is a maximum, and vice versa.

In medium 2

1 1 1 120 01

1 1

( ) ( ) (1 )j z j z j z j zi iE EH z y e e y e eβ β β β

η η− −= −Γ = −Γ

Γ 1( )H z

1( )E z

2

2

0

02

( )

( )

j zt i

j zt i

E z x E e

H z y E e

β

β

ττη

−

−

⎧ =⎪⎨

=⎪⎩



1 1

1 1

*

22 20

11

22 220

1

20

1

1 Re

2

( ) Re (1 )(1 )2

Re (1 ) ( )2

Re2

av

j z j ziav

j z j zi

i

P E H

EP z e e

Ez e e

Ez

β β

β β

η

η

η

−

−

⎡ ⎤= ×⎢ ⎥⎣ ⎦

⎡ ⎤= + Γ −Γ⎣ ⎦

⎡ ⎤= −Γ +Γ −⎣ ⎦

=2

2 201

1

220

22

1 2

2 21

2

(1 ) 2 sin 2 (1 )2

( ) ( )2

( ) ( ) for lossless medium

1

i

iav

av av

Ej z z

EP z

P P

βη

τη

η τη

⎡ ⎤−Γ + Γ = −Γ⎣ ⎦

=

=

∴ −Γ =


Normal Incidence at Multiple Dielectric Interface (1)

Reflection occur at both z=0 and z=dTotal electric field in medium 1

1 11 0 0( )j z j z

i rE x E e E eβ β−= +



Er0 is no longer related to Ei0 by Er0=ΓEi0. owing to the existence of a second discontinuity at z=d.The total reflected wave is the result of the initial reflected component and an infinite sequence of multiply reflected contributions within medium 2 that are transmitted back into medium 1.How to find the relation between Er0 and Ei0?One way is to write down the electric and magnetic field intensity vectors in all three regions and apply the boundary conditions.magnetic field in region 1

the electric and magnetic fields in region 2

1 11 0 0

1

1( ) ( )j z j z

i rH z y E e E eβ β

η−= −

2 2

2 2

2 2 2

2 2 22

( )

1( )

j z j z

j z j z

E x E e E e

H y E e E e

β β

β β

η

− ++ −

− ++ −

= +

= −



In region 3

Four unknown,Boundary conditions at two interface give birth to four equations

3

3

3 3

3 33

1

j z

j z

E xE e

H y E e

β

β

η

−+

−+

=

=

0 2 2 3, , , rE E E E+ − +

1 2

1 2

2 3

2 3

(0) (0)At 0,

(0) (0)

( ) ( )At ,

( ) ( )

E Ez

H H

E d E dz d

H d H d

⎧ =⎪= ⎨=⎪⎩

⎧ =⎪= ⎨=⎪⎩



wave impedance of the total field at any plane parallel to the plane boundary

wave impedance of the field, Z(z) : the ratio of the total electric field intensity to the total magnetic field intensity for a z-dependent uniform plane wave.

For the normal incidence of z-dependent uniform plane wave,

( )( ) ( )

( )x

y

Total E zZ z

Total H z= Ω

1 1

1 1

1 0

01

1

( ) ( )

( ) ( )

j z j zx i

j z j ziy

E z E e e

EH z e e

β β

β β

η

−

−

= + Γ

= −Γ

1 1

1 1

11 1

1

( )( ) : function of z

( )

j z j zx

j z j zy

E z e eZ z

H z e e

β β

β βη−

−

+ Γ∴ = =

−Γ



Impedance transformation

The total field in medium 2 is the result of multiple reflections of the two boundary planes at z=0 and z=d; but it can be grouped into a wave traveling in the +z direction and another traveling in the –z direction.

1 1

1 1

1 2 1 1 11 1 1

1 1 1 2 1

2 1 1 2 11

1 2 1 2 1

at z ,

( ) cos sin ( )

( ) cos sin

tan or with

tan

j l j lx

j l j ly

l

E l l j le eZ l

H l e e l j l

j l

j l

β β

β β

η β η βη ηη β η β

η η β η ηηη η β η η

−

−

= −

− ++Γ− = = =

− −Γ +

+ −= Γ =

+ +

2 1 1) If 0 and 1, then ( ) tan

wave impedance of standing wave to the left of perfect conducting

boundary

cf Z l j lη η β= Γ = − − =



The wave impedance of the total field in medium 2 at z=0

Note)As far as the wave in medium 1 is concerned, it encounters adiscontinuity at z=0 and the discontinuity can be characterized by an infinite medium with an intrinsic impedance Z2(0)

2 2

2 2

3 22 2 2 2

3 2

22 2

2

( ) ( ),

( ) ( )

j z j zx

j z j zy

E z E e e

EH z e e

β β

β β

η ηη η

η

−+

+−

−= + Γ Γ =

+

= −Γ

2 2

2 2

2 22 2

2 2

3 2 2 22 2

2 2 3 2

( )( )

( )

cos sin (0)

cos sin

j z j zx

j z j zy

E z e eZ z

H z e e

d j dZ

d j d

β β

β βη

η β η βηη β η β

− −

− −

+ Γ∴ = =

−Γ

+=

+



The effective reflection coefficient at z=0 for the incident wave in medium 1 is

Effect of transform to Z2(0) : inserting a dielectric layer of thickness d and in front of medium 3Given and , can be adjusted by suitable choices of and d

In many applications, and Er0 are the only quantities of interest; hence this impedance transform approach is conceptually simple and yields the desired answers in a direct manner.

If the fields in medium 2 and 3 are also desired, they can be determined by boundary conditions at z=0 and z=d

0 0 2 10

0 0 2 1

(0)

(0)r r

i i

E H Z

E H Z

ηη

−Γ = = − =

+

3η2η

1η 3η 0Γ 2η

0Γ

2 2, and tE E E+ −



Eg) No reflection condition

1 1 1, ,ε μ η 2η 2 2 3, ,ε μ η

d

0 2 1

3 2 2 22 1

2 2 3 2

3 2 1 222 2 1 3 2

The condition of no reflection at interface 0

0 or (0)

cos sin

cos sin

cos cos ....

sin sin ....

z

Z

d j d

d j d

d d

d d

ηη β η βη ηη β η β

η β η βη β ηη β

=⇒ Γ = =

+⇒ =

+

=⎡⎢ =⎣



3 1 2

3 1 2 3 1

22

2

or cos 0

if , can be satisfied when (trivial case of no discontinuities)

or sin 0 or 2

if cos 0, i.e) (2 1

d

nd d

d d n

η η βη η η η η

λβ

β

⇒ = == = =

= =

= = + 2

2 2 1 3

) , 0,1,2,3,..4

sin 0 and can be satisfied when

n

d

λ

β η ηη

=

≠ =

21 3

In summary, two possible conditions

1. when , , 0,1,2,...2

The thickness of the dielectric layer can be a multiple of a half-wavelength

in the dielectric layer Half-wave dielectri

d n nλη η= = =

⇒

21 3 2 1 3

c window (Narrow band device)

2. when , we require , (2 1)4

Quarter-wave impedance transformer

d nλη η η ηη≠ = = +


Oblique Incidence on the Dielectric Boundary (1)

1. TE polarizationDescription of the E-field component Ey

Asssume that wave propagates along z’ axis with dependence

r

'z

z

'x

'x

x

j te ω

'( ', ) Re ( ') , where ( ')j t j zy yE z t E z e E z Aeω β−⎡ ⎤= =⎣ ⎦



Assuming displacement vector from origin to any point on a constant phase plane

r

(x,y,z) coordinate for interface

(x',y,z') coordinate for propagation of wave

z' is constant over any plane perpendicular to ' plane wave

the perpendicular distance ' from origin to the consta

z

z

β

⇒⇒

⇒nt phase plane

' 'z z r= i

Vector wave-number : '

Then phase of plane wave : '

z

z r

β β

β β

=

= i

θ

x

z

X’

xβ

zβ

'zβ β=

Z’



Boundary condition

, where sin , cosx z x zx z

r xx y y zz

β β β β β θ β β θ= + = =

= + +

( )

'

( , ) x z

x z

j x zy

z r x z

E x y Ae β β

β β β β− +

∴ = = +

∴ =

i

0 0β ω με=E

H

βθ0, Zβ

rβ t

β1 1, Zβ

0 1β ω μ ε=

tθ



Boundary condition→Tangential component of E-field at boundary should be continuous

( ,0 ) ( ,0 ) : independent of xy yE x E x− +=

( )

( )

( )1 1

( , ) , where sin , cos



x z

r rx z

t tx z

j x ziy x z

j x zr r ry x r z r

j x zt t ty x t z t

E x z Ae

E x z Be

E x z Ce

β β

β β

β β

β β θ β β θ

β β θ β β θ

β β θ β β θ

− +

− −

− +

= = =

= = = −

= = =

( ,0 ) ( ,0 ) ( ,0 )

( ,0 ) ( ,0 )

i ry y y

ty y

E x E x E x

E x E x

− − −

+ +

= +

=

regardless of x

, : Snell's law

r tx x xj x j x j x

r tx x x

Ae Be Ce

A B C x x x

β β β

β β β

− − −+ =

∴ + = = =



If not, change of x leads to the situation that each term be out of phase

, can not have y component since and must lie in the plane

of incidence (Remember B.C.)

r t r t

r tx x x

β β β β

β β ββ

= =

1 0 0 0 1

1 1 1

1

sin sin

sin sin sin sin

sin ,

sin

r r

t t

t p

nv cn

v n u

θ β θ θ θ

β θ β θ ω μ ε θ ω μ ε θ

ε βθθ ε β

= ⇒ =

= ⇒ =

∴ = = = = =

, r tx x xA B C β β β+ = = =

FasterLess dense

SlowerMore dense

1ε ε<

θtθ

tθ θ>

FasterLess dense

SlowerMore dense

tθ

tθ θ>θ



1

1

1

Critical angle

90 , sin 1 sin

sin for , sin sin 1

sin

is complex angle

This means no refracted wave. i.e) The incident wave is total

t t c

c tc

t

v

v

βθ θ θβ

β θθ θ θ θβ θ

θ

= ° = ∴ = =

> = = >

∴

⇒

c

t

11 1 11

ly reflected

The angle of incidence which corresponds to the threshold of

total reflection , is called the critical angle.2

sin for or sinc c

n n

n n

θπθ

εθ ε ε θε

−

∴

=

⎛ ⎞= = > = ⎜⎝

⎟⎠


Oblique Incidence at a Plane Dielectric Boundary (7)

Line AO, O’A’, O’B : the intersections of the wavefronts of the incident, reflected and transmitted waves respectively, on the plane of incidence



since the phase velocity of the incident and the reflected wave are the same.

' 'OA AO=

'sin 'sin : Snell's law of reflectionr i i rOO OOθ θ θ θ= ⇒ =

1 21sin sinsin

1 1 2

) Boundary condition should be satisfied independent of x

= =

sin sin sin

i trj x j xj x

i r t

cf

e e e

x x x

β θ β θβ θ

β θ β θ β θ

− −−⇒∴ = =

2

2 1 1

2 1 1

1 2 2

'sin' ,

' 'sin

sin , cf) , : Refraction index

sin

pt

p p pi

pt

i p p p

uOOOB AO OB

u u uAO OO

u n cn

u n u u

θθ

θ β ωβθ β

= ∴ = =

= = = = =



Note)for non-magnetic media,

→A plane wave incident oblique at an interface with a denser medium will be bent toward the normal.→Snell’s law are independent of polarization

Total reflectionFor : the wave in medium1 is incident on a less dense medium 2

1 2 0μ μ μ= =

1 1 2

2 2 1

sin

sint

i

n

n

θ ε ηθ ε η

= = =

1 2ε ε>

t iθ θ>



For , : the refracted wave will be grazing along the interface2

: No refracted wave

the incident wave is totally reflected

i c t

i c

πθ θ θ

θ θ

= =

>

⇒

11 2 2

2 1 1

The angle of incidence , which corresponds to the threshold of total

reflection , is called the critical angle.2

sin let , sin or sin

2 sin

c

t

tt c c

c

n

n

θπθ

θ ε επθ θ θθ ε ε

−

∴

=

⎛ ⎞= = ⇒ = = ⎜ ⎟

⎝ ⎠



Perpendicular polarization (TE polarization)

1

1

( sin cos )0

( sin cos )0

1

Incident field

( , )

( , ) ( cos sin )

i i

i i

j x zi i

j x zii i i

E x z yE e

EH x z x z e

β θ θ

β θ θθ θη

− +

− +

=

= − +



1

1

2

( sin cos )0

( sin cos )0

1

( sin cos )0

0

Reflected field

( , )

( , ) ( cos sin )

Transmitted field into medium 2

( , )

( , )

r r

r r

t t

j x zr r

j x zrr r r

j x zt t

tt

E x z yE e

EH x z x z e

E x z yE e

EH x z

β θ θ

β θ θ

β θ θ

θ θη

− −

− −

− +

=

= +

=

= 2 ( sin cos )

2

( cos sin ) t tj x zt tx z e β θ θθ θ

η− +− +

1 21

0 0 r t

sin sinsin0 0 0

Unknowns : , , ,

Boundary conditions at the boundary 0

( ,0) ( ,0) ( ,0)

i tr

r t

iy ry ty

j jji r t

E E

z

E x E x E x

E e E e E eβ θ β θβ θ

θ θ

− −−

=+ =

⇒ + =



1 21sin sinsin 00 0

1 2

1 1 2

t 1 1

i 2 2

( ,0) ( ,0) ( ,0)

1( cos cos ) cos

Phase matching

sin sin sin

sin . ) ,

sin

Amplitu

i tr

ix rx tx

j jj ti i r r r

i r t

r c

H x H x H x

EE e E e e

x x x

ni e

n

β θ β θβ θθ θ θη η

β θ β θ β θθ βθ θθ β

− −−

+ =

⇒ − + = −

∴= =

= = =

00 0 0 0 0

1 2

de matching

1 , ( ) cos cosr

i r t i r i t

EE E E E E θ θ

η η+ = − =



2 1

0 2 1

2 10 2 1

2

0 2

2 10 2 1

Reflection coefficient

cos cos cos cos

cos coscos cos

22 cos cos

cos cos

cos cos

r i t t i

i i t

t i

t i t

i i t

t i

E

E

E

E

η ηη θ η θ θ θ

η ηη θ η θθ θη

η θ θτ η ηη θ η θθ θ

⊥

⊥

−−

Γ = = =+ +

= = =+ +

Normal Incidence

Oblique Incidence(Perpendicular)

2 1

2 1

η ηη η

−Γ =

+

2 1

2 1

cos cos

cos cos

t i

t i

η ηθ θ

η ηθ θ

⊥

−Γ =

+

1η 1

cos i

ηθ

2η2

cos t

ηθ



1 2 1 2

1

2

1 2 0 1 2

1)sin for , (very very rare case)

1

usually ,

For the most of the case there is no Brewester angle

for the TE( ) polarization

Bcf θ ε ε μ μμμ

μ μ μ ε ε

⊥ = = ≠+

= = ≠

∴⊥

22 221 1 2

2 1 1 2 1 222 1 1 20 0

1 2

2

1

Brewster angle : Incident angle for which the reflection coefficient be zero.

cos cos , where cos 1 sin , , , ,

1

sin

B t t ip

B

n cn n

n u

τ

μ ε μ μη θ η θ θ θ η ηε εμ ε

μ εμθ

⊥ ⊥

⊥

⊥

+ Γ =

= = − = = = =

−∴ = 2 1

2

1

2

1

ε

μμ

⎛ ⎞− ⎜ ⎟⎝ ⎠



Parallel Polarization (TM polarization)

1

1

( sin cos )0

( sin cos )0

1

Incident field

( , ) ( cos sin )

( , )

i i

i i

j x zi i i i

j x zii

E x z E x z e

EH x z y e

β θ θ

β θ θ

θ θ

η

− +

− +

= −

=



1

1

2

( sin cos )0

( sin cos )0

1

( sin cos )0

Reflected field

( , ) ( cos sin )

( , )

Transmitted field

( , ) ( cos sin )

( , )

r r

r r

t t

j x zr r r r

j x zrr

j x zt t t t

t

E x z E x z e

EH x z y e

E x z E x z e

H x z y

β θ θ

β θ θ

β θ θ

θ θ

η

θ θ

− −

− −

− +

= +

= −

= −

= 2 ( sin cos )0

2

t tj x ztEe β θ θ

η− +

0 0 0 0 0 01 2

Boundary condition at 0

1 1 ( ) cos cos , ( )

) The Snell's law as TE applies for TM

i r i t t i r r

z

E E E E E E

cf

θ θη η

=

+ = − =



0 2 1

0 2 1

0 2

0 2 1

Reflection coefficient

cos cos

cos cos

2 cos

cos cos

cos 1 (1+ if 0)

cos

r t i

i t i

t i

i t i

ti

i

E

E

E

E

η θ η θη θ η θ

η θτη θ η θ

θτ τ θθ

−Γ = =

+

= =+

⎛ ⎞∴ +Γ = Γ ≠ ≠⎜ ⎟

⎝ ⎠

22

Note)

example : polaroid sunglasses to reduce the sun glare

cf) E-field in parallel to the Earth's surface is predominantly reaching the eye.

Design sunglasses to filter out this

⊥Γ ≥ Γ ⇒

component.



221

2 1 22

2 1

2 1 22

1

2

221 2 1 2

2 11

2

1 12 2

1 1

Brewster angle

cos cos , cos 1 sin

1

sin

1

1 For many cases, sin for

1

or tan tan

t B t B

B

B

B

n

n

n

n

η θ η θ θ θ

μ εμ εθεε

εμ μ θ μ μ

ε εεε

εθε

− −

= = −

−=

⎛ ⎞− ⎜ ⎟⎝ ⎠

= ∴ = = =+⎛ ⎞

+ ⎜ ⎟⎝ ⎠

⎛= = ⎜

⎝1 2 for μ μ

⎞=⎟

⎠


Home work

.

8-28, 8-30, 8-32, 8-33, 8-36, 8-42

H W

field and wave electromagneticocw.snu.ac.kr/sites/default/files/note/1001.pdf · 2018-01-30 ·...

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