eleg 648 plane waves ii mark mirotznik, ph.d. associate professor the university of delaware email:...

ELEG 648Plane waves II

Mark Mirotznik, Ph.D.Associate Professor

The University of DelawareEmail: [email protected]

Uniform Plane Waves: Propagation in Any Arbitrary Direction

E

H

y

x

z

zzyyxxo

zyx

zzyyxx

zjyjxjo

EaEaEaE

zayaxar

aaa

eeeErE zyx

ˆˆˆ

ˆˆˆ

ˆˆˆ

)(~

222zyx

222

22

)sin(zyx

yx

22)cos(

yx

y


E

H

y

x

z

zzyyxxo

zyx

zzyyxx

rjo

EaEaEaE

zayaxar

aaa

eErE

ˆˆˆ

ˆˆˆ

ˆˆˆ

)(~

222zyx

222

22

)sin(zyx

yx

22)cos(

yx

y


0

)(~

o

rjo

E

eErE

E

H

y

x

z

Since E and are at right anglesfrom each other.

EaEa

EaE

EeE

Eejj

Eej

eEj

Ej

H

rjo

orj

orj

rjo

ˆ1

ˆ

ˆ1

11

11

1~1~

where 222

ˆˆˆˆ

zyx

zzyyxx aaaa

and


aaa

aaH

aaE

aErtE

trHEarH

aErtEtrEeErE

HE

Ho

Eo

Hoo

o

Eoorj

o

00

00

)cos(),(1

)(~

)cos(),()(~

Observation 1. E, H and vectors are pointing in orthogonal directions.

Summary and Observations:Frequency Domain Time Domain

Observation 2. E and H are in phase with each other, however, H’s magnitude is smaller by the amount of the wave impedance

Uniform Plane Waves: Propagation in 2D

E

H

x

y

22

)sin(

)cos(

)(~

yx

y

x

yjxjo

yx eeErE

Can we write this a bit more compact?


E

H

x

y

22

)sin(

)cos(

)(~

yx

y

x

yjxjo

yx eeErE

Can we write this a bit more compact?

yyxxo

yx

yyxx

rjo

EaEaE

yaxar

aa

eErE

ˆˆ

ˆˆ

ˆˆ

)(~


E

H

x

y

What about the polarization of E?

yyxxo

yx

yyxx

rjo

EaEaE

yaxar

aa

eErE

ˆˆ

ˆˆ

ˆˆ

)(~


E

H

x

y


0

)(~

o

rjo

E

eErE

yyxxo

yx

yyxx

rjo

EaEaE

yaxar

aa

eErE

ˆˆ

ˆˆ

ˆˆ

)(~


E

H

x

y


)cos(ˆ)sin(ˆ

)(~

oyoxo

rjo

EaEaE

eErE

Two cases

E

H

x

y

Parallel Polarization Perpendicular Polarization

ozo

rjo

EaE

eErE

ˆ

)(~


E

H

x

y

What about H?

o

o

o

o

o

o

rjo

rjo

H

H

E

E

H

E

eHrH

eErE

0

0

)(~

)(~

yyxxo

yx

yyxx

rjo

EaEaE

yaxar

aa

eErE

ˆˆ

ˆˆ

ˆˆ

)(~

o

o

o

EH


E

H

x

y

What about the polarization of H?

)cos(ˆ)sin(ˆ

)(~

oyoxo

rjo

HaHaH

eHrH

Two cases

E

H

x

y

Parallel Polarization Perpendicular Polarization

ozo

rjo

HaH

eHrH

ˆ

)(~

Reflection and Transmission

Write down the electric fields in the two regions(2 unknowns, R and T)


Next find the magnetic fields in each region


Apply boundary conditions


Write down the E field in both regions(4 unknowns, R, T, r and t)


Find the H field in both regions


Apply boundary conditions

2 equations and 4 unknowns

We need two more equations. How do we get them?

Angle of Incidence, Degrees

Ref

lect

ion

Coe

ffic

ient

Example: Reflection from an Ocean Interface

Reflection and Transmission from Dielectric Slabs

1. Normal Incidence

111

1

11

11,

222

2

22

22 ,

333

3

33

33 ,

z=0 z=d

Region I Region II Region III


Region I:

zjoRy

zjoincy

zjo

Rx

zjo

incx

eRE

H

eE

H

eERE

eEE

1

1

1

1

1

1

Region II:

zjzjIIy

zjzjIIx

eAeAH

eAeAE

22

22

212

21

1

Region III:

zjoTy

zjo

Tx

eTE

H

eETE

3

3

3

Boundary Conditions

z=0 z=d

000

000

z

IIyz

Ryz

incy

z

IIxz

Rxz

incx

HHH

EEE

dz

Tydz

IIy

dz

Txdz

IIx

HH

EE


Boundary Conditions

z=0 z=d

2

2

2

1

11

21

AAREE

AAREE

oo

oo

djodjdj

djo

djdj

eTE

eAeA

eTEeAeA

322

322

321

2

21

1

Four equations and four unknownsSolution for the Reflection Coefficient:

23

2323

12

1212

22312

22312

2

2

1

dj

dj

e

eR


Special Cases

23

2323

12

12122

2312

22312

2

2

1

dj

dj

e

eR

I. Half Wavelength Thickness Slab

""0

1

22

,2,1,0,

231231

2312

2312

2

2

2

2

reflectionnoR

thenif

R

nnn

d

nnd

o

111

1

11

11,

222

2

22

22 ,

z=0 z=2/2


111

1

11

11,


Special Cases

23

2323

12

12122

2312

22312

2

2

1

dj

dj

e

eR

II. Quarter Wavelength Thickness Slab

""0

1

442

2

2312312

2312

2312

2

2

2

2

reflectionnoR

thenif

R

d

d

o

111

1

11

11,

222

132

22 ,

z=0 z=2/4


333

3

33

33 ,

Reflection and Transmission from Dielectric Slabs: Example

23

2323

12

12122

2312

22312

2

2

1

dj

dj

e

eR

377

,

01

11

o

oo

92

222

132

2

2

10776.5

5.217

3

o

o

z=0 z=m


67.125

,9

03

33

o

oo

268.05.21767.125

5.21767.125268.0

3775.217

3775.2172312

9

9

2

2

10776.5

10776.5

22312

22312

0716.01

)1(267.0

1

j

j

dj

dj

e

e

e

eR


9

9

2

2

10776.5

10776.5

22312

22312

0716.01

)1(267.0

1

j

j

dj

dj

e

e

e

eR

Frequency, MHz

|R|

How do we broaden the bandwidth around the zero reflection point?

Frequency, MHz

|R|

One Solution is Multiple Dielectric Layers

111

1

11

11,

222

2

22

22 ,

333

3

33

33 ,

4

4

5

5

6

6


1. Oblique Incidence ( Parallel Polarization)

111

1

11

11,

222

2

22

22 ,

333

3

33

33 ,

z=0 z=d


i

t

r II

II


Region I:

])sin(1)cos(1[])sin(1)cos(1[


])sin(1)cos(1[

11

11

11

])sin()cos([

)sin(ˆ)cos(ˆ~

)sin(ˆ)cos(ˆ~

ˆ~

ˆ~

yizijyizij

yizijyizij

yizij

ii

eRE

aeRE

aH

eE

aeE

aH

eREaE

eEaE

oizi

oy

R

oizi

oy

inc

oxR

yzjox

inc

Region II:

zy

zyII

yzjyzjIIx

aaeA

aaeA

H

eAeAE

yzj

yzj

ˆ)sin(ˆ)cos(

ˆ)sin(ˆ)cos(~

222

2

222

1

])sin()cos([2

])sin()cos([1

])2sin(2)2cos(2[

])2sin(2)2cos(2[

22222222


Region III:


])sin(3)cos(3[

33

3

)sin(ˆ)cos(ˆ~

ˆ~

ytztjytztj

ytztj

eTE

aeTE

aH

eTEaE

oiz

oy

T

oxT

Boundary Conditions

z=0 z=d

000

000

z

IIyz

Ryz

incy

z

IIxz

Rxz

incx

HHH

EEE

dz

Tydz

IIy

dz

Txdz

IIx

HH

EE

Phase Matching Conditions

)sin()sin(

)sin()sin(

322

221

t

i


Six Equations and Six Unknowns

)sin()sin(

)sin()sin(

322

221

t

i

dtjdjdj

dtjdjdj

eT

EeA

eA

TeEeAeA

AAREE

AAREE

to

o

oo

oo

)cos(3)2cos(2)2cos(2

)cos(3)2cos(2)2cos(2

)cos()cos()cos(

)cos()cos(

32

2

22

2

1

21

22

22

2

1

11

21

Reflection and Transmission from Dielectric Slabs: Solution (parallel polarization)

dj

dj

e

eR

2

2

22312

22312

1

)cos()cos(

)cos()cos(

)cos()cos(

)cos()cos(

))sin((sin

))sin((sin

322

32223

221

22112

23

21

2

112

22

t

t

i

i

t

i

o

*note we have assumed all non-magneticmaterials here

Reflection and Transmission from Dielectric Slabs: Solution (perpendicular polarization)

dj

dj

e

eR

2

2

22312

22312

1

)cos()cos(

)cos()cos(

)cos()cos(

)cos()cos(

))sin((sin

))sin((sin

223

22323

212

21212

23

21

2

112

22

t

t

i

i

t

i

o

*note we have assumed all non-magneticmaterials here


377

,

01

11

o

oo

92

222

132

2

2

10776.5

5.217

3

o

o

z=0 z=m

Region I Region IIRegion III

67.125

,9

03

33

o

oo

eleg 648 plane waves ii mark mirotznik, ph.d. associate professor the university of delaware email:...

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